Additive structure of totally positive quadratic integers

TL;DR

This paper characterizes the additive structure of totally positive integers in real quadratic fields, identifying generators, relations, and uniquely decomposable integers, and shows this structure uniquely determines the field.

Contribution

It provides a complete description of the additive semigroup of totally positive integers in real quadratic fields using continued fractions, and proves this structure uniquely identifies the field.

Findings

Generators and relations described via continued fractions

Characterization of uniquely decomposable integers

Semigroup determines the quadratic field

Abstract

Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and determine its generators (indecomposable integers) and relations; they can be nicely described in terms of the periodic continued fraction for $\sqrt D$. We also characterize all uniquely decomposable integers in $K$ and estimate their norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$ completely determines the real quadratic field $K$.

Equations

$${\omega }_D:=\{\begin{array}{ll}\sqrt{D}& \text{if }D\equiv 2,3(\mathrm{mod}4),\\ \frac{1+\sqrt{D}}{2}& \text{if }D\equiv 1(\mathrm{mod}4).\end{array}$$

$$\sigma_{D}\coloneqq\omega_{D}+\lfloor-\omega_{D}^{\prime}\rfloor=\begin{cases}\sqrt{D}+\lfloor\sqrt{D}\rfloor&\text{if }D\equiv 2,3\mathclose{}\ (\operatorname{mod}4),\\ \frac{1+\sqrt{D}}{2}+\big{\lfloor}\frac{-1+\sqrt{D}}{2}\big{\rfloor}&\text{if }D\equiv 1\mathclose{}\ (\operatorname{mod}4)\end{cases}$$

$$X_{i+2}=u_{i+2}{X}_{i+1}+X_i\text{for}i\ge -1$$

$$N_i:=|N({\alpha }_i)|=(-1{)}^{i+1}N({\alpha }_i)=\{\begin{array}{ll}|p_i^2-D{q}_i^2|& \text{if }D\equiv 2,3(\mathrm{mod}4),\\ |p_i^2-p_i{q}_i-q_i^2\frac{D-1}{4}|& \text{if }D\equiv 1(\mathrm{mod}4).\end{array}$$

$$\mathcal{A}:=\{{\alpha }_{i,r}:i\ge -1\text{ odd and }0\le r\le u_{i+2}-1\}\backslash \{1\}$$

$${\mathcal{O}}_K$$

$${\alpha }_{i,r-1}-2{\alpha }_{i,r}+{\alpha }_{i,r+1}=0\text{for odd i≥−1 and 1≤r≤ui+2−1},$$

$${\alpha }_{i-2,u_i-1}-(u_{i+1}+2){\alpha }_{i,0}+{\alpha }_{i,1}=0\text{for odd i≥1},$$

$$\text{same relations as in eqrefeq:R1 and eqrefeq:R2 after applying the automorphism (′)},$$

$${\alpha }_{-1,1}^{\prime }-(u_0+2)\cdot 1+{\alpha }_{-1,1}=0.$$

$$v_j{\beta }_j={\beta }_{j-1}+{\beta }_{j+1},$$

$$v_j:=\{\begin{array}{ll}2& \text{if β∣j∣=αi,r with odd i≥−1 and 1≤r≤ui+2−1},\\ u_{i+1}+2& \text{if β∣j∣=αi,0 with odd i≥−1}.\end{array}$$

$$\begin{gathered} {\alpha }_{i-2,u_i-1}+{\alpha }_{i,1}={\alpha }_{i-2}+(u_i-1){\alpha }_{i-1}+{\alpha }_i+{\alpha }_{i+1} \\ ={\alpha }_i-{\alpha }_{i-1}+{\alpha }_i+u_{i+1}{\alpha }_i+{\alpha }_{i-1}=(u_{i+1}+2){\alpha }_i=(u_{i+1}+2){\alpha }_{i,0}. \end{gathered}$$

$${\beta }_1+{\beta }_{-1}=\mathrm{Tr}({\beta }_1)=\mathrm{Tr}(\lceil u_0/2\rceil -{\omega }_D^{\prime }+1)=2\lceil u_0/2\rceil -\mathrm{Tr}({\omega }_D)+2=u_0+2.\text{qed}$$

$${\beta }_{j-1}-v_j{\beta }_j+{\beta }_{j+1}=0\text{for}j\in \mathbb{Z}.$$

$$\frac{{\beta }_{j_0}}{{\beta }_{j_0}^{\prime }}\le \frac{x}{x^{\prime }}<\frac{{\beta }_{j_0+1}}{{\beta }_{j_0+1}^{\prime }}.$$

$$x=e{\beta }_{j_0}+f{\beta }_{j_0+1}\text{with}e,f\in \mathbb{Z}$$

$$x=(k_{j_{\min }+1}+v_{j_{\min }+1}{k}_{j_{\min }}){\beta }_{j_{\min }+1}+(k_{j_{\min }+2}-k_{j_{\min }}){\beta }_{j_{\min }+2}+\sum _{j=j_{\min }+3}^{j_{\max }}{k}_j{\beta }_j$$

$$x=\sum _{j=j_{\min }}^{j_{\max }-3}{k}_j{\beta }_j+(k_{j_{\max }-2}-k_{j_{\max }}){\beta }_{j_{\max }-2}+(k_{j_{\max }-1}+v_{j_{\max }-1}{k}_{j_{\max }}){\beta }_{j_{\max }-1}$$

$$0\le f({\beta }_{j_0+1}{\beta }_{j_0}^{\prime }-{\beta }_{j_0}{\beta }_{j_0+1}^{\prime })\text{and}0<e({\beta }_{j_0+1}{\beta }_{j_0}^{\prime }-{\beta }_{j_0}{\beta }_{j_0+1}^{\prime }).$$

$$\frac{{\beta }_{j_1}}{{\beta }_{j_1}^{\prime }}<\frac{{\beta }_{j_1+1}}{{\beta }_{j_1+1}^{\prime }}\le \frac{x}{x^{\prime }}=\frac{e_1{\beta }_{j_1}+f_1{\beta }_{j_1+1}}{e_1{\beta }_{j_1}^{\prime }+f_1{\beta }_{j_1+1}^{\prime }}.$$

$$\sum _{h_j>0}{h}_j{\beta }_j-\sum _{h_j<0}(-h_j){\beta }_j=0,\text{i.e.,}x:=\sum _{h_j>0}{h}_j{\beta }_j=\sum _{h_j<0}(-h_j){\beta }_j.$$

$$\mathcal{T}:B_{j-1}+B_{j+1}=v_j{B}_j\text{for}j\in \mathbb{Z}$$

$$1\le e\le v_j-1,0\le f\le v_{j+1}-1\text{and}(e,f)\ne (v_j-1,v_{j+1}-1).$$

$$x+{\beta }_j⪯v_j{\beta }_j+v_{j+1}{\beta }_{j+1}\text{or}x+{\beta }_{j+1}⪯v_j{\beta }_j+v_{j+1}{\beta }_{j+1}.$$

$$\sum _{i=1}^s{u}_i+2\sum _{\begin{array}{c}i=2\\ i\text{ even}\end{array}}^s{u}_i+\sum _{\begin{array}{c}i=1\\ i\text{ odd},u_i=1\end{array}}^{s-1}{u}_{i-1}{u}_{i+1}$$

$$4\sum _{i=1}^s{u}_i+\sum _{\begin{array}{c}i=1\\ u_i=1\end{array}}^s{u}_{i-1}{u}_{i+1}$$

$$\text{(a) }\sum u_{i+2};\text{(b) }\sum u_{i+1};\text{(c) }\sum _{u_{i+2}\ge 2}{u}_{i+3};\text{(d) }\sum _{u_{i+2}\ge 2}{u}_{i+1};$$

$$\displaystyle\text{(e) }\sum_{\mathclap{u_{i+2}=1}}\bigl{(}(u_{i+1}+1)(u_{i+3}+1)-1\bigr{)}=\sum_{\mathclap{u_{i+2}=1}}(u_{i+1}+u_{i+3})+\sum_{\mathclap{u_{i+2}=1}}u_{i+1}u_{i+3},$$

$$A:=\{\alpha \in S:\text{α indecomposable and 2α uniquely decomposable}\}.$$

$$k_{\alpha }\alpha +\beta \text{ is uniquely decomposable},$$

$$B\coloneqq\big{\{}\,\{\alpha-\beta,\beta-\alpha\}\mathclose{}:\mathopen{}\alpha\in A,\,\alpha,\beta\text{ satisfy~{}\eqref{eq:kab}}\,\big{\}},$$

$$\dots ,{\alpha }_3^{\prime },\{{\alpha }_2^{\prime },-{\alpha }_2^{\prime }\},{\alpha }_1^{\prime },\{{\alpha }_0^{\prime },-{\alpha }_0^{\prime }\},{\alpha }_{-1}={\alpha }_{-1}^{\prime }=1,\{{\alpha }_0,-{\alpha }_0\},{\alpha }_1,\{{\alpha }_2,-{\alpha }_2\},{\alpha }_3,\dots$$

$$\dots ,u_{s-1},u_s=u_0,u_1,u_2,\dots ,u_{s-2},u_{s-1},u_s=u_0,u_1,\dots$$

Full text

We acknowledge support by Czech Science Foundation (GAČR) grant 17-04703Y (TH,VK) and partial support by Charles University Research Centre program UNCE/SCI/022 (VK)

1. Introduction

The additive semigroup of totally positive integers $\mathcal{O}_{K}^{+}$ in a totally real number field $K$ has long played a fundamental role in algebraic number theory, even though more attention has perhaps been paid to the multiplicative structure of the ring $\mathcal{O}_{K}$, for example, to its units and unique factorization into primes. The most prominent purely additive objects are the indecomposable elements, i.e., totally positive integers $\alpha\in\mathcal{O}_{K}^{+}$ that cannot be decomposed into a sum $\alpha=\beta+\gamma$ of totally positive integers $\beta,\gamma\in\mathcal{O}_{K}^{+}$. For example, in 1945, Siegel used them (under the name “extremal elements”) to prove that if $K$ is a number field different from $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{5})$, then there is a totally positive integer in $K$ that cannot be written as sum of any number of squares [Sie45].

In the real quadratic case $K=\mathbb{Q}(\sqrt{D})$, indecomposables can be nicely characterized in terms of continued fraction (semi-)convergents to $\sqrt{D}$ (see Section 2); Dress and Scharlau [DS82] proved an upper bound on the norm of each indecomposable $N(\alpha)\leq D$, which was recently refined by Jang, Kim, Tinková, Voutier, and the second author [JK16, Kal16a, TV18]. This stands in contrast to the situation of a general totally real field $K$, where it is much harder to describe indecomposables: Brunotte [Bru83] proved an upper bound on their norm in terms of the regulator, but otherwise their structure remains quite mysterious.

The goal of this article is to study the structure of the whole additive semigroup $\mathcal{O}_{K}^{+}(+)$. This is an interesting problem in itself, but it seems also necessary for certain applications (such as the recent progress in the study of universal quadratic forms and lattices over $K$ by Kim, Blomer, Yatsyna, and the second author [Kim00, BK15, BK18, Kal16b, KY18, Yat19]).

In particular, as indecomposable elements are precisely the generators of $\mathcal{O}_{K}^{+}(+)$, we need to determine the relations between them. While the description of indecomposables in terms of the continued fraction is fairly straightforward, it is a priori not clear at all if the same will be the case for relations, as there could be some “random” or “accidental” ones. Perhaps surprisingly, it turns out that this is not the case and that all the relations (given in Theorem 2) can be described quite elegantly. A key tool in the proof is the fact that each totally positive integer can be uniquely written as a $\mathbb{Z}^{+}$-linear combination of two consecutive indecomposables. We also show in Corollary 3 that this corresponds to a cancellative presentation of the semigroup $\mathcal{O}_{K}^{+}(+)$.

One of course cannot hope to have an analogue of unique factorization in the additive setting, but nevertheless, some elements can be uniquely decomposed as a sum of indecomposables. In Theorem 4 we characterize all such uniquely decomposable elements and obtain again a very explicit result depending only on the continued fraction. As another application, this then yields a direct proof of Theorem 6 that $\mathcal{O}_{K}^{+}(+)$ (viewed as an abstract semigroup) completely determines $D$ and the number field $K$. Let us briefly remark that this result can be viewed alongside a number of beautiful results concerning the (im)possibility of reconstructing a number field from some of its invariants, such as the absolute Galois group, Dedekind zeta-function, or Dirichlet $L$-series, see, e.g., [Gaß26, Kub57, Neu69, Uch76, CdSL*+*19]. A natural question to ask is of course whether an analogue of our Theorem 6 holds also for totally real number fields of higher degree, although this may be quite hard, as we lack a good understanding of indecomposable elements.

Finally, we use our results to estimate the norms of totally positive integers, and in particular, analogously to the results that norms of convergents and indecomposables are at most $2D^{1/2}$ and $D$, respectively, we show that the norm of a uniquely decomposable elements is at most of the order $D^{3/2}$ (Theorem 10).

Note that structure theory of semigroups is a well-developed subject, e.g., see [CP67, Gri95], and the references therein. Among the fundamental topics in the area are classification problem (characterize a given class of semigroups, e.g., by finding their presentations) and isomorphism problem (decide when two presentations give isomorphic semigroups): our Corollary 3 and Theorem 6 solve these problems for the semigroups $\mathcal{O}_{K}^{+}(+)$ (which are not finitely generated). Semigroups also have important applications to theoretical computer science, for example in relation to the study of formal languages. Quite similar to our semigroups $\mathcal{O}_{K}^{+}(+)$, that can be viewed as subsets of $\mathbb{Z}^{2}$, are the important linear and semilinear subsets of $\mathbb{Z}^{n}$ [Gin66, Ch. 5]). Also in number theory, numerical semigroups, i.e., subsemigroups of $\mathbb{Z}^{+}(+)$, are commonly studied, e.g., [GSR99] study their presentations. As a final example, let us mention that additive subsemigroups of $\mathbb{Z}^{n}$, that have similar geometric properties as $\mathcal{O}_{K}^{+}$, were recently used to classify a certain class of semifields [KK18].

Acknowledgments

We are grateful to the anonymous referee for pointing out that we were using the incorrect notion of semigroup presentation and for several other very useful comments that helped us improve the article.

2. Preliminaries

Throughout the work, we will use the following notation. We fix a squarefree integer $D\geq 2$ and consider the real quadratic field $K=\mathbb{Q}(\sqrt{D})$ and its ring of integers $\mathcal{O}_{K}$; we know that $\{1,\omega_{D}\}$ forms an integral basis of $\mathcal{O}_{K}$, where

[TABLE]

By $\Delta$ we denote the discriminant of $K$, i.e., $\Delta=4D$ if $D\equiv 2,3\mathclose{}\ (\operatorname{mod}4)$ and $\Delta=D$ otherwise. The norm and trace from $K$ to $\mathbb{Q}$ are denoted by $N$ and $\operatorname{Tr}$, respectively.

An algebraic integer $\alpha\in\mathcal{O}_{K}$ is totally positive iff $\alpha>0$ and $\alpha^{\prime}>0$, where $\alpha^{\prime}$ is the Galois conjugate of $\alpha$, we write this fact as $\alpha\succ 0$; for $\alpha,\beta\in\mathcal{O}_{K}$ we denote by $\alpha\succ\beta$ the fact that $\alpha-\beta\succ 0$, and by $\mathcal{O}_{K}^{+}$ the set of all totally positive integers. We say that $\alpha\in\mathcal{O}_{K}^{+}$ is indecomposable iff it can not be written as a sum of two totally positive integers or equivalently iff there is no algebraic integer $\beta\in\mathcal{O}_{K}^{+}$ such that $\alpha\succ\beta$. We say that $\alpha\in\mathcal{O}_{K}^{+}$ is uniquely decomposable iff there is a unique way how to express it as a sum of indecomposable elements.

It will be slightly more convenient for us to work with a purely periodic continued fraction, and so let $\sigma_{D}=[\overline{u_{0},u_{1},\dots,u_{s-1}}]$ be the periodic continued fraction expansion of

[TABLE]

(with positive integers $u_{i}$). We then have that \omega_{D}=\bigl{[}\lceil u_{0}/2\rceil,\overline{u_{1},\dots,u_{s}}\bigr{]}. It is well known that $u_{1},u_{2},\dots,u_{s-1}$ is a palindrome and that $u_{0}=u_{s}$ is even if and only if $D\equiv 2,3\bmod 4$, hence $\lceil u_{0}/2\rceil=(u_{s}+\operatorname{Tr}(\omega_{D}))/2$ [Per13, § 24.II].

Denote the convergents to $\omega_{D}$ by p_{i}/q_{i}\coloneqq\bigl{[}\lceil u_{0}/2\rceil,u_{1},\dots,u_{i}\bigr{]} and recall that the sequences $(p_{i})$, $(q_{i})$ satisfy the recurrence

[TABLE]

with the initial condition $q_{-1}=0$, $p_{-1}=q_{0}=1$, and $p_{0}=\lceil u_{0}/2\rceil$ [Per13, § 1]. Denote $\alpha_{i}\coloneqq p_{i}-q_{i}\omega_{D}^{\prime}$ and $\alpha_{i,r}=\alpha_{i}+r\alpha_{i+1}$. Then we have the following classical facts (see, e.g., [DS82]):

• •

The sequence $(\alpha_{i})$ satisfies the recurrence (1).

• •

We have that $\alpha_{i}\succ 0$ if and only if $i\geq-1$ is odd.

• •

The indecomposable elements in $\mathcal{O}_{K}^{+}$ are $\alpha_{i,r}$ with odd $i\geq-1$ and $0\leq r\leq u_{i+2}-1$, together with their conjugates.

• •

We have that $\alpha_{i,u_{i+2}}=\alpha_{i+2,0}$.

• •

The indecomposables $\alpha_{i,r}$ are increasing with increasing $(i,r)$ (in the lexicographic sense).

• •

The indecomposables $\alpha_{i,r}^{\prime}$ are decreasing with increasing $(i,r)$.

We also denote $\varepsilon>1$ the fundamental unit of $\mathcal{O}_{K}$; we have that $\varepsilon=\alpha_{s-1}$. Furthermore, we denote $\varepsilon^{+}>1$ the smallest totally positive unit $>1$; we have that $\varepsilon^{+}=\varepsilon$ if $s$ is even and $\varepsilon^{+}=\varepsilon^{2}=\alpha_{2s-1}$ if $s$ is odd. Furthermore, we denote $\gamma_{0}=\omega_{D}$ and $\gamma_{i}=[u_{i},u_{i+1},u_{i+2},\dotsc]$ for $i\geq 1$; we have that $u_{i}<\gamma_{i}=u_{i}+\frac{1}{\gamma_{i+1}}<u_{i}+1$ for $i\geq 1$.

In the final Section 5 we will estimate norms of totally positive integers and particularly uniquely decomposable integers. To this end, we will use the following additional notation: For a convergent $\alpha_{i}$, we set

[TABLE]

Recall that we have $p_{i+1}q_{i}-p_{i}q_{i+1}=(-1)^{i}$ and let us define $T_{i}$ so that $\alpha_{i}\alpha_{i+1}^{\prime}=T_{i+1}+(-1)^{i}\omega_{D}$. This means that we have $T_{i}=p_{i}(p_{i-1}-q_{i-1})-q_{i}q_{i-1}\frac{D-1}{4}$ or $T_{i}=p_{i}p_{i-1}-Dq_{i}q_{i-1}$ when $D\equiv 1\pmod{4}$ or $2,3\pmod{4}$, respectively.

3. Relations in the semigroup $\mathcal{O}_{K}^{+}(+)$

In this section we will consider all the (additive) relations that hold in the semigroup $\mathcal{O}_{K}^{+}(+)$. We will show in Theorem 2 that they all follow from certain basic relations, and that this amounts to giving a presentation of $\mathcal{O}_{K}^{+}(+)$ as a commutative, cancellative semigroup (Corollary 3).

Let

[TABLE]

denote the set of indecomposable elements $>1$ and $\mathcal{A}^{\prime}\coloneqq\{\,y^{\prime}\mathclose{}:\mathopen{}y\in\mathcal{A}\,\}$ the set of their conjugates, that is, of indecomposables $<1$. Note that each totally positive integer is a finite sum of indecomposables (for, by considering the trace $\operatorname{Tr}\colon K\rightarrow\mathbb{Q}$ it is easy to see that there are no infinite descending chains in $\mathcal{O}_{K}^{+}$ with respect to $\succ$); in other words, the indecomposables $\mathcal{A}\cup\mathcal{A}^{\prime}\cup\{1\}$ generate the semigroup $\mathcal{O}_{K}^{+}(+)$.

[TABLE]

We will shortly see that these relations hold (in fact, it is a straightforward verification from the definitions), but let us first introduce an alternative notation of the indecomposables for convenience. We define $\beta_{j}$, $j\in\mathbb{Z}$, by the condition that $\dotsb<\beta_{-3}<\beta_{-2}<\beta_{-1}<\beta_{0}=1<\beta_{1}<\beta_{2}<\beta_{3}<\dotsb$ is the increasing sequence of the indecomposables. Note that we have $\beta_{j}^{\prime}=\beta_{-j}$ for all $j\in\mathbb{Z}$.

Lemma 1**.**

For each $j\in\mathbb{Z}$ we have that

[TABLE]

where

[TABLE]

Proof.

As $\beta_{-j}=\beta_{j}^{\prime}$, we can assume $j\geq 0$. We have $\beta_{j}=\alpha_{i,r}$ for some odd $i\geq-1$ and $0\leq r\leq u_{i+2}-1$.

If $1\leq r\leq u_{i+2}-1$, then $\beta_{j-1}=\alpha_{i,r-1}=\alpha_{i}+(r-1)\alpha_{i+1}$, $\beta_{j}=\alpha_{i,r}=\alpha_{i}+r\alpha_{i+1}$ and $\beta_{j+1}=\alpha_{i,r+1}=\alpha_{i}+(r+1)\alpha_{i+1}$ form an arithmetic sequence, whence $\beta_{j}-\beta_{j-1}=\beta_{j+1}-\beta_{j}$, which is the statement.

If $r=0$ and $i\geq 1$, we have a multiple of $\alpha_{i,0}=\alpha_{i}$ as the left-hand side and $\alpha_{i-2,u_{i}-1}+\alpha_{i,1}$ as the right-hand side. We use the definition of $\alpha_{i,r}$ and the recurrence (1) for $\alpha_{j}$ to see that

[TABLE]

Finally, consider $r=0$ and $i=-1$, i.e., the case $j=0$. We have $\beta_{1}=\alpha_{-1,1}=\alpha_{1}+1=\lceil u_{0}/2\rceil-\omega^{\prime}+1$, whence

[TABLE]

Note that with the notation from Lemma 1, we can rewrite the relations (2a)–(2d) in a unified way in terms of $\beta_{j}$ and $v_{j}$ as follows:

[TABLE]

As our first main theorem, we can now show that each totally positive integer is a linear combination of two consecutive indecomposables with non-negative integral coefficients. A variant of this statement concerning sums of powers of units was used by Kim, Blomer, and the second author [Kim00, BK18] in the construction of universal quadratic forms.

Theorem 2**.**

Let $x\in\mathcal{O}_{K}^{+}$ be given as a finite sum $x=\sum k_{j}\beta_{j}$ with $k_{j}\in\mathbb{Z}$. Then there exist unique $j_{0},e,f\in\mathbb{Z}$ with $e\geq 1$ and $f\geq 0$ such that $x=e\beta_{j_{0}}+f\beta_{j_{0}+1}$.

Every relation of the form $\sum h_{j}\beta_{j}=0$ (with $h_{j}\in\mathbb{Z}$ and only finitely many non-zeros) is a $\mathbb{Z}$-linear combination of the relations (3); in particular, this is true for $e\beta_{j_{0}}+f\beta_{j_{0}+1}-\sum k_{j}\beta_{j}=0$.

Proof.

As the sequence $(\beta_{j})_{j\in\mathbb{Z}}$ is strictly increasing from [math] to $+\infty$ and the sequence $(\beta_{j}^{\prime})_{j\in\mathbb{Z}}$ is strictly decreasing from $+\infty$ to [math], we know that the sequence $(\beta_{j}/\beta_{j}^{\prime})_{j\in\mathbb{Z}}$ is strictly increasing from [math] to $+\infty$. As $x\succ 0$, we have that $x/x^{\prime}>0$. Hence we can fix the unique index $j_{0}$ such that

[TABLE]

We will first show that $x=\sum_{j=j_{\min}}^{j_{\max}}k_{j}\beta_{j}$ can be rewritten to

[TABLE]

using only relations (3). We assume that $j_{\min}\leq j_{0}$ and $j_{\max}\geq j_{0}+1$ (if not, we pad the sum by zeros to achieve this). We use induction on the length of the sum $L=j_{\max}-j_{\min}+1\geq 2$:

• •

Suppose $L=2$. Then $j_{\min}=j_{0}$ and $j_{\max}=j_{0}+1$. Putting $e=k_{j_{0}}$ and $f=k_{j_{0}+1}$ gives the statement.

• •

Suppose $L\geq 3$ and $j_{\min}\leq j_{0}-1$. Then

[TABLE]

(here we used (3) for $j=j_{\min}+1$) is a sum of the same form with length $L-1$.

• •

Suppose $L\geq 3$ and $j_{\min}=j_{0}$. Then $j_{\max}=j_{0}+L-1\geq j_{0}+2$ and

[TABLE]

(here we used (3) for $j=j_{\max}-1$) is a sum of the same form with length $L-1$.

This finishes the proof of (5) (note that so far we have not used the property of $j_{0}$).

We now plug (5) into (4) to derive that

[TABLE]

As $\beta_{j_{0}}^{\prime}>\beta_{j_{0}+1}^{\prime}>0$ and $\beta_{j_{0}+1}>\beta_{j_{0}}>0$, we have that $\beta_{j_{0}+1}\beta_{j_{0}}^{\prime}-\beta_{j_{0}}\beta_{j_{0}+1}^{\prime}>0$, hence $e>0$ and $f\geq 0$.

To prove the uniqueness of $j_{0}$ from the statement of the proposition, suppose to the contrary that $j_{1}\neq j_{0}$ and $x=e_{1}\beta_{j_{1}}+f_{1}\beta_{j_{1}+1}$ with $e_{1}>0$ and $f_{1}\geq 0$. We first consider the case when $j_{1}\leq j_{0}-1$, i.e.,

[TABLE]

Easy manipulation leads to $e_{1}(\beta_{j_{1}+1}\beta_{j_{1}}^{\prime}-\beta_{j_{1}}\beta_{j_{1}+1}^{\prime})\leq 0$, hence $e_{1}\leq 0$. Likewise, in the case $j_{1}\geq j_{0}+1$ we get $f_{1}<0$.

Once $j_{0}$ is fixed, the uniqueness of $e,f$ follows from the linear independence of $\beta_{j_{0}},\beta_{j_{0}+1}$ over $\mathbb{Q}$.

For the second part of the theorem, note that we have already proved this for $e\beta_{j_{0}}+f\beta_{j_{0}+1}-\sum k_{j}\beta_{j}=0$ (as we deduced (5) only using (3)).

Consider now any relation between indecomposables $\sum h_{j}\beta_{j}=0$ (where of course only finitely many $h_{j}$’s are non-zero). Let us write this as two sums,

[TABLE]

Clearly $x\in\mathcal{O}_{K}^{+}$. By the first part of the theorem, $x$ is uniquely written as $x=e\beta_{j}+f\beta_{j+1}$ for some $j\in\mathbb{Z}$, $e\geq 1$, $f\geq 0$; and we know that both $e\beta_{j}+f\beta_{j+1}=\sum_{h_{j}>0}h_{j}\beta_{j}$ and $e\beta_{j}+f\beta_{j+1}=\sum_{h_{j}<0}(-h_{j})\beta_{j}$ follow from (3). Thus also the identity between the two right-hand sides follows from (3). ∎

In order to show that Theorem 2 amounts to a certain presentation of the semigroup $\mathcal{O}_{K}^{+}(+)$, let us first recall some relevant definitions (see [Gri95] for more background and for any undefined notions). Note that we will use the additive notation, as we are interested in the additive semigroup $\mathcal{O}_{K}^{+}(+)$, although in the literature it is perhaps more common to view semigroups multiplicatively.

Let $S(+)$ be a commutative semigroup, i.e., a set $S$ together with a binary operation $+$ that is commutative and associative. $S$ is cancellative iff $x+z=y+z$ implies $x=y$ (for all $x,y,z\in S$). In this case, let $G(S)=S-S$ be the universal (or Grothendieck) group of $S$, i.e., the group of (formal) differences $a-b$ for $a,b\in S$, where $a-b=c-d$ in $G(S)$ iff $a+d=b+c$ in $S$ (see [Gri95, Ch. II, §§ 1, 2]). As $S$ is cancellative, we can view it as a subset of $G(S)$, i.e., $S\subset G(S)$.

Let us now introduce presentations (following, e.g., [Gri95, Ch. I]). Let $X$ be a set and let $\mathcal{R}$ be a set of relations between the elements of $X$, i.e., formal expressions $x_{1}+\dots+x_{u}=y_{1}+\dots+y_{v}$ for $x_{i},y_{j}\in X$. Let $F_{X}(+)$ be the free commutative semigroup over the set of generators $X$ and let $\mathcal{C}$ be the congruence on $F_{X}$ generated by the relations $\mathcal{R}$ (see [Gri95, Ch. 1, Proposition 2.9]). Then $\langle X\mid\mathcal{R}\rangle$ is a presentation of a commutative semigroup $S(+)$ iff $S$ is isomorphic to the quotient $F_{X}/\mathcal{C}$ of $F_{X}$ by the congruence $\mathcal{C}$. Note that $F_{X}/\mathcal{C}$ in general need not be cancellative (although $F_{X}$ is), and so, when considering cancellative semigroups, it is sometimes more convenient to consider cancellative presentations.

We say that $\langle X\mid\mathcal{R}\rangle_{\mathrm{c}}$ is a cancellative presentation of a cancellative, commutative semigroup $S(+)$ iff $S$ is isomorphic to the quotient $F_{X}/\mathcal{D}$, where $\mathcal{D}$ is the smallest cancellative congruence on $F_{X}$ that contains the relations $R$. The universal property of presentations implies that it is also isomorphic to the factor of $\langle X\mid\mathcal{R}\rangle$ by the smallest cancellative congruence (cf. [Gri95, Ch. II, Proposition 2.3]).

A little less formally, $\langle X\mid\mathcal{R}\rangle_{\mathrm{c}}$ is a cancellative presentation a cancellative, commutative semigroup $S(+)$ if $S$ is generated by $X$ and all (additive) relations between elements of $X$ are generated by consecutively applying the relations in $\mathcal{R}$ and by using the cancellative law ‘$x+z=y+z$ implies $x=y$’.

Corollary 3**.**

Let $X=\{\,B_{j}\mathclose{}:\mathopen{}j\in\mathbb{Z}\,\}$ and

[TABLE]

(where $v_{j}$ are as in Lemma 1). Then $\langle X\mid\mathcal{T}\rangle_{\mathrm{c}}$ is a cancellative presentation of the cancellative, commutative semigroup $\mathcal{O}_{K}^{+}(+)$.

Proof.

Let us consider the (semigroup) homomorphism $\varphi\colon F_{X}\rightarrow\mathcal{O}_{K}^{+}$ defined by $\varphi(B_{j})=\beta_{j}$.

The indecomposables $\beta_{j}$ are generators of $\mathcal{O}_{K}^{+}$, and so $\varphi$ is surjective. Let $\mathcal{E}$ be the corresponding congruence on $F_{X}$, i.e., $a\mathcal{E}b$ iff $\varphi(a)=\varphi(b)$. Hence we have an isomorphism of cancellative semigroups $\mathcal{O}_{K}^{+}\simeq F_{X}/\mathcal{E}$ and $\mathcal{E}$ is a cancellative congruence.

Let further $\mathcal{D}$ be the smallest cancellative congruence on $F_{X}$ that contains the relations $\mathcal{T}$. Since the relations (corresponding to) $\mathcal{T}$ hold for $\beta_{j}$ by Lemma 1, we have $\mathcal{T}\subset\mathcal{E}$. Furthermore, as $\mathcal{O}_{K}^{+}$ is a cancellative commutative semigroup, we have $\mathcal{D}\subset\mathcal{E}$. In order to show that $\langle X\mid\mathcal{T}\rangle_{\mathrm{c}}$ is indeed cancellative presentation of $\mathcal{O}_{K}^{+}(+)$, it remains to show that $\mathcal{D}=\mathcal{E}$. This is essentially clear from Theorem 2, but we can also proceed formally as follows.

As in [Gri95, Ch. II, Proposition 5.1], let $R(\mathcal{D})$ be the Rédei group of $\mathcal{D}$, i.e., $R(\mathcal{D})$ is the subgroup of $G(F_{X})$ (which is clearly the free commutative group generated by $X$) such that $R(\mathcal{D})=\{\,a-b\mathclose{}:\mathopen{}a,b\in F_{X},a\mathcal{D}b\,\}$. Similarly let let $R(\mathcal{E})$ be the Rédei group of $\mathcal{E}$. The Rédei group uniquely determines the corresponding cancellative congruence, and so it suffices to show that $R(\mathcal{D})=R(\mathcal{E})$.

By the minimality of $\mathcal{D}$, we see that $R(\mathcal{D})$ is the smallest subgroup of $G(F_{X})$ that contains $B_{j-1}-v_{j}B_{j}+B_{j+1}$ for all $j$. By definition, $R(\mathcal{E})$ consists precisely of sums $\sum h_{j}B_{j}$ (with $h_{j}\in\mathbb{Z}$) such that $\sum h_{j}\beta_{j}=0$ holds (in $\mathcal{O}_{K}$). But by the second part of Theorem 2, each such relation $\sum h_{j}\beta_{j}=0$ follows from the relations (3). We have thus proved that $R(\mathcal{D})=R(\mathcal{E})$. ∎

Note that it is easy to see that the given set of relations (7) is minimal in the sense that none of them can be removed.

Let us also note that we indeed need to work with cancellative presentations. For example, $B_{0}+B_{3}=(v_{1}-1)B_{1}+(v_{2}-1)B_{2}$ holds in $\langle X\mid\mathcal{T}\rangle_{\mathrm{c}}$, but does not hold in $\langle X\mid\mathcal{T}\rangle$, as we cannot apply any of the relations (7) to this identity (without using cancellation).

4. Uniquely decomposable elements

We now characterize all the uniquely decomposable elements in $\mathcal{O}_{K}^{+}$ using the results of the previous section.

Theorem 4**.**

All uniquely decomposable elements $x\in\mathcal{O}_{K}^{+}$ are the following:

1. (a)

$\alpha_{i,r}$* with odd $i\geq-1$ and $0\leq r\leq u_{i+2}-1$;* 2. (b)

$e\alpha_{i,0}$* with odd $i\geq-1$ and with $2\leq e\leq u_{i+1}+1$* 3. (c)

$\alpha_{i,u_{i+2}-1}+f\alpha_{i+2,0}$* with odd $i\geq-1$ odd such that $u_{i+2}\geq 2$ and with $1\leq f\leq u_{i+3}$;* 4. (d)

$e\alpha_{i,0}+\alpha_{i,1}$* with odd $i\geq-1$ such that $u_{i+2}\geq 2$ and with $1\leq e\leq u_{i+1}$;* 5. (e)

$e\alpha_{i,0}+f\alpha_{i+2,0}$* with odd $i\geq-1$ such that $u_{i+2}=1$ and with $1\leq e\leq u_{i+1}+1$, $1\leq f\leq u_{i+3}+1$, $(e,f)\neq(u_{i+1}+1,u_{i+3}+1)$;* 6. (f)

Galois conjugates of all of the above.

Note that the first item lists all the indecomposables (which are clearly uniquely decomposable) and the second one comprises small multiples of all convergents $\alpha_{i}\succ 0$, including positive integers $1,2,\dots,u_{0}+1$.

Proof.

The statement of the theorem, when rewritten using $\beta_{j}$ and $v_{j}$ (defined in Lemma 1), says that $x$ is uniquely decomposable if and only if $x=e\beta_{j}+f\beta_{j+1}$ with

[TABLE]

Let $x$ be uniquely decomposable. By Theorem 2, it can be written as $x=e\beta_{j}+f\beta_{j+1}$ for some $e,f,j\in\mathbb{Z}$ with $e\geq 1,f\geq 0$. Suppose now that $e,f$ do not satisfy (8). This can happen only in one of the following ways:

• •

$e\geq v_{j}$. Then $x=e\beta_{j}+f\beta_{j+1}=\beta_{j-1}+(e-v_{j})\beta_{j}+(f+1)\beta_{j+1}$ are two decompositions of $x$ (see Lemma 1).

• •

$f\geq v_{j+1}$. Then $x=e\beta_{j}+f\beta_{j+1}=(e+1)\beta_{j}+(f-v_{j+1})\beta_{j+1}+\beta_{j+2}$ are two decompositions of $x$.

• •

$(e,f)=(v_{j}-1,v_{j+1}-1)$. Then $x=(v_{j}-1)\beta_{j}+(v_{j+1}-1)\beta_{j+1}=\beta_{j-1}+\beta_{j+2}$ are two decompositions of $x$.

Conversely, let us now show that the condition (8) is sufficient. First note that (8) implies

[TABLE]

Suppose now that $x=e\beta_{j}+f\beta_{j+1}$ with $e,f$ satisfying (8) is not uniquely decomposable. This can happen only in one of the following ways:

• •

$x$ has a decomposition that contains some $\beta_{i}$ with $i\geq j+2$, i.e., $x\succeq\beta_{i}\geq\beta_{j+2}$. Simultaneously, we have from (9) that $x+\beta_{j-1}\leq v_{j}\beta_{j}+v_{j+1}\beta_{j+1}=\beta_{j-1}+\beta_{j+2}$, whence $x<\beta_{j+2}$. This is a contradiction.

• •

$x$ has a decomposition that contains some $\beta_{i}$ with $i\leq j-1$, i.e., $x\succeq\beta_{i}$. Then $x^{\prime}\geq\beta_{i}^{\prime}\geq\beta_{j-1}^{\prime}$. We also have from (9) that $x^{\prime}+\beta_{j+2}^{\prime}\leq v_{j}\beta_{j}^{\prime}+v_{j+1}\beta_{j+1}^{\prime}=\beta_{j-1}^{\prime}+\beta_{j+2}^{\prime}$, and so $x^{\prime}<\beta_{j-1}^{\prime}$, a contradiction.

• •

$x=e\beta_{j}+f\beta_{j+1}=e_{1}\beta_{j}+f_{1}\beta_{j+1}$ for $(e,f)\neq(e_{1},f_{1})$. This is impossible as $\beta_{j},\beta_{j+1}$ are linearly independent over $\mathbb{Q}$.

∎

Note that similarly to the classical formula for the number of indecomposables as a sum of some coefficients $u_{i}$ (e.g., [BK15]), we can now express the number of uniquely decomposable elements.

Corollary 5**.**

The number of uniquely decomposable elements of $\mathcal{O}_{K}^{+}$ modulo totally positive units (i.e., powers of $\varepsilon^{+}$) is equal to

[TABLE]

Proof.

Denote $s^{+}=s$ for $s$ even and $s^{+}=2s$ for $s$ odd. Then by direct computation from Theorem 4, we obtain that the number is, for each item in the theorem statement:

[TABLE]

where all the sums are over odd $i$ between $1$ and $s^{+}$, because $\alpha_{i+s^{+}}=\varepsilon^{+}\alpha_{i}$ for all odd $i\geq-1$ so this restriction on $i$ picks one representative from the uniquely decomposable integers for each class modulo powers of $\varepsilon^{+}$. Rearranging the sums and using that $u_{i+s}=u_{i}$ we get the result $\sum_{i=1}^{s^{+}}u_{i}+2\sum_{\begin{subarray}{c}\\ i=1\\ \mathclap{i\text{ even\vphantom{d}}}\end{subarray}}^{s^{+}}u_{i}+\sum_{\begin{subarray}{c}\\ i=1\\ \mathclap{i\text{ odd},\,u_{i}=1}\end{subarray}}^{s^{+}-1}u_{i-1}u_{i+1}$; this finishes the proof for $s$ even.

For $s$ odd, note that all the summands are perodic with period $s$, hence a sum over all $1\leq i\leq s^{+}$ is twice the sum over all $1\leq i\leq s$ and a sum over odd $1\leq i\leq s^{+}$ is equal to the sum over all $1\leq i\leq s$. ∎

As another application, we use Theorem 4 to prove that the additive semigroups of totally positive integers of different real quadratic fields are non-isomorphic. This is in stark contrast to the situation of the groups $\mathcal{O}_{K}(+)$, which are all isomorphic to $\mathbb{Z}^{2}(+)$.

Theorem 6**.**

The additive semigroups $\mathcal{O}_{K}^{+}$, for real quadratic fields $K$, are pairwise not isomorphic.

Proof.

Assume that we are given $\mathcal{O}_{K}^{+}(+)$ as an abstract semigroup $S(+)$. To prove the uniqueness of $K=\mathbb{Q}(\sqrt{D})$, we shall reconstruct the continued fraction for $\sigma_{D}$ using the additive structure of $S$. Note that the indecomposable and uniquely decomposable elements are well-defined in $S$ as they are defined intrinsically just from the semigroup structure. Since $S$ is a cancellative semigroup, we can consider the universal (or Grothendieck) group of differences $G(S)=S-S$ and have $S\subset G(S)$.

Consider all the indecomposables such that their double is uniquely decomposable:

[TABLE]

From Theorem 4 we know that $A$ is exactly the set of convergents and their conjugates, A=\{\,\alpha_{i},\alpha_{i}^{\prime}\mathclose{}:\mathopen{}\text{i odd}\,\}. For $\alpha\in A$, denote $k_{\alpha}$ the maximum integer such that $k_{\alpha}\alpha$ is uniquely decomposable. From Theorem 4cd we know that for each $\alpha\in A$ there are exactly two $\beta\in S$ such that

[TABLE]

namely if $\alpha=\alpha_{i}$ then $\beta=\alpha_{i,1}=\alpha+\alpha_{i+1}$ or $\beta=\alpha_{i-2,u_{i}-1}=\alpha_{i-2,u_{i}}-\alpha_{i-1}=\alpha-\alpha_{i-1}$.

Consider now an infinite (bipartite) graph $G$ with vertices $A\cup B$, where

[TABLE]

where $\alpha-\beta\in G(S)$. Note that $B$ actually contains pairs $\{\alpha_{i+1},-\alpha_{i+1}\}$ and $\{\alpha_{i+1}^{\prime},\allowbreak-\alpha_{i+1}^{\prime}\}$ with odd $i$ (but we have no intrinsic way of distinguishing $\alpha_{i+1}$ from $-\alpha_{i+1}$). The edges in $G$ are defined as follows: There is an edge between $\alpha\in A$ and $\{\gamma,-\gamma\}\in B$ iff $\beta=\alpha-\gamma$ or $\beta=\alpha+\gamma$ satisfy (10). Then $G$ is actually an infinite chain corresponding to

[TABLE]

We add labels on each vertex of $G$ in the following way: We label $\alpha\in A$ by $k_{\alpha}-1$. For $\{\gamma,-\gamma\}\in B$ and its neighbors $\alpha,\tilde{\alpha}$, we have that $\alpha-\tilde{\alpha}=l_{\gamma}\gamma$ for some $l_{\gamma}\in\mathbb{Z}$ and we label $\{\gamma,-\gamma\}$ by $\lvert l_{\gamma}\rvert$. To see the motivation behind this, assume that $\gamma=\alpha_{i+1}$ for some odd $i$; then $\alpha-\tilde{\alpha}=\pm(\alpha_{i+2}-\alpha_{i})=\pm(\alpha_{i}+u_{i+2}\alpha_{i+1}-\alpha_{i})=\pm u_{i+2}\alpha_{i+1}$.

Whence we know that for $i$ odd, $\alpha_{i}$ is labelled by $u_{i+1}$ and $\{\alpha_{i+1,}-\alpha_{i+1}\}$ is labelled by $u_{i+2}$ (with the same being true for $\alpha_{i}^{\prime}$). This means that the infinite chain of the labels is equal to

[TABLE]

(where it does not matter in which direction we read the chain as it is a palindrome). We know that $u_{0}$ is the maximal value in the chain and that $s$ is the (shortest) period of the chain. This means that we have reconstructed $\sigma_{D}=[\overline{u_{0};u_{1},\dots,u_{s-1}}]$ just from the intrinsic properties of $S(+)$. ∎

5. Estimating norms

We have seen in Theorem 2 that (up to conjugation) every element of $\mathcal{O}_{K}^{+}$ can be uniquely written in the form $e\alpha_{i,r}+f\alpha_{i,r+1}$, where $i\geq-1$ is odd, $0\leq r\leq u_{i+2}-1$, $e\geq 1$, and $f\geq 0$. We will now estimate the norm of such an element, generalizing the results concerning indecomposables by Dress, Scharlau, Jang, Kim, the second author (and others) [DS82, JK16, Kal16a]; here we will use the notation introduced at the end of Section 2.

Let us start by recalling the following classical fact (for the proof see, e.g., [Kal16a, Proposition 5] and [BK18, Lemma 3]).

Lemma 7**.**

For all $i\geq-1$, we have

[TABLE]

We are interested in

[TABLE]

and so let us first prove a general formula for the norm of the element on the right-hand side.

Lemma 8**.**

Let $m,n\in\mathbb{Z}$ and $i\geq-1$ odd. Then

[TABLE]

Proof.

Let’s prove the result only when $D\equiv 2,3\pmod{4}$, as the other case is very similar. Using the definitions and the previous lemma, we compute

[TABLE]

Now we return to the original situation when $m=e+f$ and $n=re+(r\mkern 2.0mu{+}\mkern 2.0mu1)f$ and estimate the norm of totally positive integers.

Proposition 9**.**

Consider $\alpha=e\alpha_{i,r}+f\alpha_{i,r+1}\in\mathcal{O}_{K}^{+}$ with $i\geq-1$ odd, $0\leq r<u_{i+2}$, $e\geq 1$, and $f\geq 0$. Then we have the following upper bounds on $N(\alpha)$:

[TABLE]

For a lower bound we distinguish four cases:

1. (1)

If $f>0$ and $r=0$, then $N(\alpha)>ef{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}$. 2. (2)

Let $c\in(0,1)$. If $f>0$ and $1\leq r\leq cu_{i+2}-1$, then $N(\alpha)>(1-c)(e+f)^{2}{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}$. 3. (3)

If $f>0$ and $\frac{u_{i+2}+1}{2}<r\leq u_{i+2}-1$, then $N(\alpha)>\frac{{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}}{2}e(e+f)$. 4. (4)

If $f=0$ and $r>0$, then N(\alpha)>e^{2}\bigl{(}1-\frac{1}{u_{i+2}}\bigr{)}{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}.

Note that when $f=r=0$, then $\alpha=e\alpha_{i}$ is a multiple of a convergent and we have a good control on the size of its norm by Lemma 7. Hence we have not included the lower bound for this case in the proposition.

Proof.

Let $m=e+f$ and $n=re+(r\mkern 2.0mu{+}\mkern 2.0mu1)f$. By the previous proposition, we want to estimate

[TABLE]

We have A=e\bigl{(}1-\frac{r}{\gamma_{i+2}}\bigr{)}+f\bigl{(}1-\frac{r+1}{\gamma_{i+2}}\bigr{)} and $B=n{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+AN_{i}$.

We start with the upper bounds. We have $A<e+f$ and $B<n{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+(e+f){\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}={\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}((r\mkern 2.0mu{+}\mkern 2.0mu1)e+(r\mkern 2.0mu{+}\mkern 2.0mu2)f)$, where we used the easy consequence of Lemma 7 that $N_{i}<{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}$.

For the second estimate we use a slightly different argument (which we again give only in the case $D\equiv 2,3\pmod{4}$):

[TABLE]

We finish with the lower bounds. Let us assume first that $f>0$ and distinguish three cases according to the size of $r$.

• •

Case $r=0$. Then $A>e$ and $B>{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}f$.

• •

Case $1\leq r\leq cu_{i+2}-1$. Then $r+1\leq cu_{i+2}<c\gamma_{i+1}$, and so $1-\frac{r}{\gamma_{i+2}}>1-\frac{r+1}{\gamma_{i+2}}>1-c$. Thus A=e\bigl{(}1-\frac{r}{\gamma_{i+2}}\bigr{)}+f\bigl{(}1-\frac{r+1}{\gamma_{i+2}}\bigr{)}>(1-c)(e+f). Moreover, $B>{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}(e+f)r$.

• •

Case $\frac{u_{i+2}+1}{2}<r\leq u_{i+2}-1$. Again, $B>{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}(e+f)r$. Note that the function r\mapsto\bigl{(}1-\frac{r}{\gamma_{i+2}}\bigr{)}r is increasing for $r>\frac{u_{i+2}+1}{2}$. If we apply this observation, we obtain Ar>e\bigl{(}1-\frac{r}{\gamma_{i+2}}\bigr{)}r>e\bigl{(}1-\frac{u_{i+2}-1}{\gamma_{i+2}}\bigr{)}(u_{i+2}-1)=e\frac{1+(\gamma_{i+2}-u_{i+2})}{u_{i+2}+(\gamma_{i+2}-u_{i+2})}(u_{i+2}-1)>e\frac{u_{i+2}-1}{u_{i+2}}>\frac{e}{2}, where we use the fact that $u_{i+2}\geq r+1\geq 2$.

Finally, if $f=0$, then $N(\alpha)=e^{2}N(\alpha_{i,r})$, so proving the case $e=1$ is sufficient. If $r=0$, then estimates on the norm of the convergent $\alpha_{i}$ are well-known, see, e.g., [BK18, Lemma 5]. Assume hence $1\leq r\leq u_{i+2}-1$. We have

[TABLE]

Since the minimum of the function r\mapsto\bigl{(}1-\frac{r}{\gamma_{i+2}}\bigr{)}r is at one of the endpoints of the considered interval $1\leq r\leq u_{i+2}-1$, we conclude that N(\alpha_{i,r})>\bigl{(}1-\frac{1}{u_{i+2}}\bigr{)}{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}. ∎

Let us conclude the paper with some applications of the previous proposition. From the theory of continued fractions it quite easily follows that all elements with absolute value of norm less than ${\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}/4$ are convergents [BK18, Proposition 7]. However, often it is useful to know all elements of norm less than ${\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}$ (or other small multiples of $\sqrt{D}$): the characterization of all such totally positive integers follows easily from our proposition.

Finally, we prove an upper bound on the norm of uniquely decomposable elements, similar to the bounds on norms of convergents $N_{i+1}=|N(\alpha_{i})|<\frac{{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}}{u_{i+1}}$ and indecomposables $N(\alpha_{i,r})\leq\frac{\Delta}{4N_{i+1}}$ (for odd $i$).

Theorem 10**.**

If $\alpha\in\mathcal{O}_{K}^{+}$ is uniquely decomposable, then

[TABLE]

Proof.

All uniquely decomposable $\alpha$ are described in Theorem 4, and so we just need to check the estimate on the norm in each of these cases. This is very easy when $\alpha$ is an indecomposable or a multiple of a convergent. In the rest of the proof we will often use the easy observation that $u_{j}\leq u_{s}<{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}$ for all $j$.

Case $\alpha=e\alpha_{i}+f\alpha_{i+2}$ with odd $i\geq-1$ such that $u_{i+2}=1$ and with $1\leq e\leq u_{i+1}+1$, $1\leq f\leq u_{i+3}+1$, $(e,f)\neq(u_{i+1}+1,u_{i+3}+1)$. In the notation of Proposition 9 we have $r=0$, and so the first upper bound gives $N(\alpha)<{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}(e+2f)(e+f)$. Thus the estimate follows from the range of $e,f$ using $u_{j}<{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}$.

Case $e\alpha_{i,0}+\alpha_{i,1}$ with odd $i\geq-1$ such that $u_{i+2}\geq 2$ and with $1\leq e\leq u_{i+1}$. We have $f=1$ and $r=0$, and so we similarly as in the previous case get $N(\alpha)<{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}({\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+1)({\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+2)$.

Case $\alpha_{i,u_{i+2}-1}+f\alpha_{i+2,0}$ for $i\geq-1$ odd and $1\leq f\leq u_{i+3}$. Now $e=1$ and $r=u_{i+2}-1$. In the notation (11) we have A=\bigl{(}1-\frac{u_{i+2}-1}{\gamma_{i+2}}\bigr{)}+f\bigl{(}1-\frac{u_{i+2}}{\gamma_{i+2}}\bigr{)}<\frac{2}{u_{i+2}}+\frac{f}{u_{i+2}}=\frac{f+2}{u_{i+2}}<\frac{{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+2}{u_{i+2}}. Because $n<(f+1)(r\mkern 2.0mu{+}\mkern 2.0mu1)$, we also have $B<(f+1)u_{i+2}{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+AN_{i}<({\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+1)u_{i+2}{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+\frac{{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}+2}{u_{i+2}}{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}.$ Finally,

[TABLE]

The bound is essentially sharp in the first case discussed in the proof: For example, consider the situation when $i$ is odd, $u\coloneqq u_{i+1}=u_{i+3}>{\mathchoice{\textstyle\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}{\sqrt{\!\Delta}}}/(1+\varepsilon)$ (for a constant $\varepsilon>0$), $u_{i+2}=1$, and $\alpha=u\alpha_{i}+u\alpha_{i+2}=u(2\alpha_{i}+\alpha_{i+1})$. As $\gamma_{i+2}>1$, using Lemma 8 we get that

[TABLE]

There are infinitely many such examples for any $\varepsilon>0$, because there are infinitely many squarefree $D=n^{2}-1$, which then have $u_{2j}=2\lfloor\sqrt{D}\rfloor$ and $u_{2j+1}=1$ for all $j$. Examples with $\varepsilon\geq 4$, say, seem to be fairly common.

Note that one could also further refine the estimates in terms of sizes of the coefficients $u_{j}$, similarly as was done in [JK16] for the norms of indecomposables.