Scaling of the diffraction measure of $k$-free integers near the origin

TL;DR

This paper derives asymptotic formulas for how the diffraction intensity of $k$-free integers scales near the origin, providing insights into the structure and fluctuations of these number sets.

Contribution

It introduces new asymptotic results for the diffraction measure of $k$-free integers, advancing understanding of their local fluctuation behavior.

Findings

Derived asymptotics for diffraction intensity near the origin.

Quantified the degree of patch fluctuations in $k$-free integers.

Enhanced the mathematical understanding of the structure of $k$-free integers.

Abstract

Asymptotics are derived for the scaling of the total diffraction intensity for the set of $k$-free integers near the origin, which is a measure for the degree of patch fluctuations.

Equations

$$f_{k}(q)\,\mathrel{\mathop{:}}=\,\begin{cases}\prod_{p|\bar{q}}\frac{1}{p^{k}-1},&\mbox{if $q$ is $(k+1)$-free,}\\ 0,&\mbox{otherwise.}\end{cases}$$

$$\phi (x,q):=\mathrm{card}\{1⩽m⩽qx:\gcd (m,q)=1\}$$

$$Z_k(x)=\sum _{\begin{array}{c}q\in {\mathbb{N}}_{k+1}\\ q⩾1/x\end{array}}\phi (x,q)f_k(q{)}^2,$$

$$\log\bigl{(}Z_{k}(x)\bigr{)}\,\sim\,\left(2-\frac{\raisebox{-2.0pt}{$1$}}{\raisebox{0.5pt}{$k$}}\right)\log(x)\hskip 0.5pt.$$

$${\gamma }_k^{}=\sum _{m\in \mathbb{Z}}{\eta }_k^{}(m){\delta }_m,$$

$$L^{\circledast}_{k}\,\mathrel{\mathop{:}}=\,\big{\{}y\in\mathbb{Q}:y=\tfrac{m}{q}\text{ with }m\in\mathbb{Z},\,q\in\mathbb{N}_{k+1}\text{ and }\gcd(m,q)=1\big{\}},$$

$$\widehat{{\gamma }_k}=\sum _{z\in L_k^{⊛}}{I}_k(z){\delta }_z,$$

$$I_k(z)={\left(\frac{f_k(q)}{\zeta (k)}\right)}^2$$

$$Z_{k}(x)\,\mathrel{\mathop{:}}=\,\frac{\widehat{\hskip 0.5pt\gamma_{k}\hskip 0.5pt}\bigl{(}(0,x]\bigr{)}}{\widehat{\hskip 0.5pt\gamma_{k}\hskip 0.5pt}\bigl{(}\{0\}\bigr{)}}\,=\,\zeta(k)^{2}\!\sum_{\begin{subarray}{c}z\in L^{\circledast}_{k}\\ 0<z\leqslant x\end{subarray}}I_{k}(z)\,=\sum_{q\geqslant 1/x}\sum_{\begin{subarray}{c}1\leqslant m\leqslant qx\\ \gcd(m,q)=1\end{subarray}}f_{k}(q)^{2},$$

$$Z_k(x)=\sum _{\begin{array}{c}q\in {\mathbb{N}}_{k+1}\\ {\bar{q}}^k⩾1/x\end{array}}\phi (x,q)f_k(q{)}^2.$$

$$f_k(q)=\frac{1}{\phi (\bar{q})\sigma ({\bar{q}}^{k-1})}\text{and}\frac{1}{{\bar{q}}^k}<f_k(q)<\frac{\zeta (k)}{{\bar{q}}^k}.$$

$$\sigma ({\bar{q}}^{k-1})=\prod _{p|\bar{q}}\frac{p^k-1}{p-1}={\bar{q}}^{k-1}\prod _{p|\bar{q}}\frac{1-p^{-k}}{1-p^{-1}}.$$

$${\bar{q}^{\hskip 0.5ptk}}{f_{k}(q)}\,=\,\frac{\bar{q}^{\hskip 0.5ptk}}{\varphi(\bar{q})\hskip 0.5pt\hskip 0.5pt\sigma(\bar{q}^{\hskip 0.5ptk-1})}\,=\,\prod_{p|\bar{q}}\frac{1}{1-p^{-k}}\,\in\,\bigl{(}1,{\zeta(k)}\bigr{)},$$

$$\phi (x,q)=\phi (\ell x,\bar{q}).$$

$$\left|\phi (x,q)-x\phi (q)\right|⩽d(q).$$

$$\phi (x,q)⩽x\phi (q)+d(q).$$

$$\sum_{q\geqslant{y^{-1}}}\frac{\lvert\mu(q)\rvert}{q^{k}}\,=\,\frac{y^{k-1}}{(k-1)\hskip 0.5pt\zeta(2)}+O\bigl{(}y^{k-\frac{1}{2}}\bigr{)}\,=\,\frac{y^{k-1}}{(k-1)\hskip 0.5pt\zeta(2)}\bigl{(}1+O(y^{1/2})\bigr{)}.$$

$$\sum _{q⩾y^{-1}}\frac{|\mu (q)|}{q^k}=k{\int }_{y^{-1}}^{\infty }M(t)\frac{\mathrm{d}t}{t^{k+1}}-y^k\sum _{q<y^{-1}}|\mu (q)|.$$

$$\sum _{q⩾y^{-1}}\frac{|\mu (q)|}{q^k}$$

$$\displaystyle=\,\frac{k}{\zeta(2)}\int_{y^{-1}}^{\infty}\frac{\,\mathrm{d}t}{t^{k}}\,-\,\frac{y^{k-1}}{\zeta(2)}\,+\,O\!\left(\int_{y^{-1}}^{\infty}\frac{\,\mathrm{d}t}{t^{k+\frac{1}{2}}}\right)+O\bigl{(}y^{k-\frac{1}{2}}\bigr{)}$$

$$\displaystyle=\,\frac{y^{k-1}}{(k-1)\hskip 0.5pt\zeta(2)}\,+\,O\bigl{(}y^{k-\frac{1}{2}}\bigr{)}.\qed$$

$$Z_k(x)$$

$$⩾\sum _{q^{\alpha }⩾1/x}|\mu (q)|f_k(q{)}^2\sum _{\ell |q^{k-1}}\phi (\ell x,q)$$

$$\displaystyle=\sum_{q^{\alpha}\geqslant 1/x}|\mu(q)|\,f_{k}(q)^{2}\sum_{\ell|q^{k-1}}\Bigl{(}\ell x\hskip 0.5pt\varphi(q)+O\bigl{(}d(q)\bigr{)}\Bigr{)}.$$

$$Z_k(x)$$

$$=x\sum _{q^{\alpha }⩾1/x}|\mu (q)|f_k(q)+O\left(\sum _{q^{\alpha }⩾1/x}|\mu (q)|d(q)d(q^{k-1})f_k(q{)}^2\right)$$

$$⩾x\sum _{\begin{array}{c}q⩾x^{-1/\alpha }\end{array}}\frac{|\mu (q)|}{q^k}+O\left(\sum _{q⩾x^{-1/\alpha }}\frac{|\mu (q)|d(q)d(q^{k-1})}{q^{2k}}\right),$$

$$d(q)d(q^{k-1})⩽q^{\epsilon }.$$

$$Z_k(x)⩾x\sum _{\begin{array}{c}q⩾x^{-1/\alpha }\end{array}}\frac{|\mu (q)|}{q^k}+O\left(\sum _{q⩾x^{-1/\alpha }}\frac{|\mu (q)|}{q^{2k-\epsilon }}\right).$$

$$\sum _{q⩾x^{-1/\alpha }}\frac{|\mu (q)|}{q^{2k-\epsilon }}=O_{\epsilon }\left(x^{\frac{1}{\alpha }(2k-1-\epsilon )}\right),$$

$$Z_k(x)⩾\frac{x^{\frac{k-1}{\alpha }+1}}{(k-1)\zeta (2)}+O\left(x^{\frac{2k-1}{2\alpha }+1}\right)+O_{\epsilon }\left(x^{\frac{1}{\alpha }(2k-1-\epsilon )}\right).$$

$$\alpha <k-\epsilon .$$

$$Z_k(x)⩾\frac{x^{\left(2-\frac{1}{k}\right)\left(\frac{1}{1-\epsilon }\right)-\frac{\epsilon }{1-\epsilon }}}{(k-1)\zeta (2)}+O\left(x^{\left(2-\frac{1}{2k}\right)\left(\frac{1}{1-\epsilon }\right)-\frac{\epsilon }{1-\epsilon }}\right)+O_{\epsilon }\left(x^{\left(2-\frac{1}{k}\right)\left(\frac{1}{1-\epsilon }\right)-\frac{1}{k}\left(\frac{\epsilon }{1-\epsilon }\right)}\right).$$

Full text

1. Introduction

Given a point set $\varLambda$ in Euclidean $d$-space, an immediate natural question to ask is that of the existence of its density, and further, if the density does exist, how it fluctuates locally. More generally, one can consider point patches and define means and variances for their appearance. When $\varLambda$ is a lattice, there is no fluctuation (in the sense that each patch repeats lattice-periodically), while a random set such as the positions of an ideal gas (as modelled by a Poisson process) shows a lot of fluctuation. What intermediate regimes exist is a natural aspect of spatial order, and such considerations have lately garnered much interest.

More recently, due to general progress in the theory of aperiodic order, see [2] and references therein for background, fluctuation or variance considerations have been extended to such systems, both in the projection realm and in the context of inflation systems [16, 17, 3]. The key to many of these investigations is understanding the scaling behaviour of the diffraction measure $\widehat{\gamma}$, which is the mathematical counterpart of the structure factor used in physics, in the vicinity of the origin (in reciprocal space). In one dimension, the scaling behaviour of Z(x)\mathrel{\mathop{:}}=\widehat{\gamma}\bigl{(}(0,x]\bigr{)}/\hskip 0.5pt\widehat{\gamma}\bigl{(}\{0\}\bigr{)} as $x\to 0^{+}$ is an important tool for this [20, 3]. Indeed, one can interpret a decay rate $Z(x)/x\to 0$, as $x\to 0^{+}$, of the diffraction intensity as fluctuations (as $R\to\infty$ in direct space) of the point set and its patches in density; see [15] for early results on aperiodic examples. Within this framework, Torquato and Stillinger [20] introduced and investigated the notion of hyperuniformity, referring to point patterns that do not possess infinite-wavelength fluctuations.

While one gets $Z(x)=x$ for the homogeneous Poisson process of unit density in $\mathbb{R}\hskip 0.5pt$, power laws of the form $Z(x)\sim x^{\alpha}$ with $\alpha>1$, as $x\to 0^{+}$, are typical for aperiodically ordered sets [15, 16, 17]; in this way, these sets are hyperuniform. Systems such as the Thue –Morse chain show a decay that is faster than any power [11, 3], and a lattice displays a function $Z(x)$ that drops down to [math] for sufficiently small $x>0$, thus signifying a perfect (in fact, periodic) repetition of patches of any size. In this way, the statistic $Z(x)$ allows for various notions of intermediate order; for a nice discussion on this topic, see the introduction of Brauchart, Grabner and Kusner [8], as well as [3] for a summary of results.

Naturally, one can also address such questions for point sets of number-theoretic origin, such as the square-free integers on the line and their various generalisations; see [4, 18, 5] and references therein for systematic examples. Here, we consider the set $V_{k}$ of $k$-free integers with fixed $k\geqslant 2$, that is, the elements of $\mathbb{Z}$ that are not divisible by the $k$-th power of any (rational) prime number. The sets $V_{k}$ are examples of weak model sets of maximal density, and are thus approachable by the projection method [5, 13]. The diffraction measure of $V_{k}$ is known to be a pure point measure [7], which is explicitly computed in terms of elementary number-theoretic functions. So, we can investigate the function $Z_{k}(x)$ for this family of point sets in $\mathbb{Z}\subset\mathbb{R}\hskip 0.5pt$. With the knowledge of the diffraction measure, which will be recalled below, our task can be termed as follows.

For any integer $q\neq 0$, let $\bar{q}\mathrel{\mathop{:}}=\textnormal{rad}(q)$ denote the square-free part of $q$, that is, its square-free divisor of largest modulus. For $q,k\in\mathbb{N}$ with $k\geqslant 2$, set

[TABLE]

Clearly, $f_{k}(1)=1$, and $f_{k}(q)=f_{k}(\bar{q})$ when $q$ is $(k+1)$-free. Also, write

[TABLE]

for the $2$-parameter totient function, where both $x>0$ and $q\in\mathbb{N}$ are assumed. Note that $\varphi(x,q)=0$ whenever $qx<1$. In this article, we are interested in analysing the asymptotics, as $x\to 0^{+}$, of

[TABLE]

where $\mathbb{N}_{j}$ with $j\geqslant 2$ denotes the set of $j$-free positive integers; that is, $\mathbb{N}_{j}=V_{j}\cap\mathbb{N}$.

As our main result, we shall establish the following asymptotic.

Theorem 1.1**.**

Let $k\geqslant 2$ be a fixed positive integer. As $x\to 0^{+}$, we have

[TABLE]

From here on, the article is organised as follows. In Section 2, we recall the essential results on the diffraction measure of the $k$-free integers and then state some results of elementary or asymptotic nature that we require for the proof of Theorem 1.1 in Section 3. Finally, we briefly comment on further developments in Section 4.

2. Preliminaries and known results

Let us begin by recalling some results from [7] on the diffraction of $k$-free integers. It is well known that the set $V_{k}$ has natural density $1/\zeta(k)$, which can be obtained as a limit along any averaging sequence of growing intervals around an arbitrary, but fixed centre. This is called tied density in [7]. If any such averaging sequence is fixed, say \bigl{(}[-n,n]\bigr{)}_{n\in\mathbb{N}} for instance, also all patch frequencies exist; in particular, the $2$-point correlations within $V_{k}$ exist. If $\eta_{k}(m)$ is the frequency of occurrence of two points within $V_{k}$ at distance $m\in\mathbb{Z}\setminus\{0\}$, and $\eta_{k}(0)=1/\zeta(k)$ is the density of $V_{k}$, one obtains the autocorrelation measure of $V_{k}$ as

[TABLE]

where $\delta_{x}$ denotes the normalised Dirac measure at $x$; compare [2, Ch. 9 and Sec. 10.4].

The autocorrelation measure is positive definite, and hence Fourier transformable as a measure [2, Prop. 8.6]; $\widehat{\hskip 0.5pt\gamma_{k}\hskip 0.5pt}$ is known as the diffraction measure of $V_{k}$. The main result of [7] on $\widehat{\hskip 0.5pt\gamma_{k}\hskip 0.5pt}$ can be summarised as follows.

Theorem 2.1**.**

The diffraction measure $\widehat{\hskip 0.5pt\gamma_{k}\hskip 0.5pt}$ of $V_{k}$ is a pure point measure, supported on

[TABLE]

which is a subgroup of $\mathbb{Q}$. More precisely, the diffraction measure reads

[TABLE]

where, for any $z=\frac{m}{q}\in L^{\circledast}_{k}$, the intensity is given by

[TABLE]

with the function $f_{k}(q)$ as defined in Eq. (1.1). ∎

Note that $0=\frac{0}{1}\in L^{\circledast}_{k}$, with I_{k}(0)=\zeta(k)^{-2}=\widehat{\hskip 0.5pt\gamma_{k}\hskip 0.5pt}\bigl{(}\{0\}\bigr{)}. Clearly, the diffraction measure is reflection symmetric with respect to the origin, which means that the scaling near [math] can be determined from the positive side. Here, for $x>0$, one gets

[TABLE]

which leads immediately to Eq. (1.3). Now, according to the definition of $\varphi(x,q)$, without changing the value of $Z_{k}(x)$, we can enlarge the summation set in (1.3) to include all $q\in\mathbb{N}_{k+1}$ subject to the weaker condition $\bar{q}^{k}\geqslant 1/x$. This is possible since, whenever $q\in\mathbb{N}_{k+1}$, we have both $q\leqslant\bar{q}^{k}$ and $\varphi(x,q)=0$ for all terms with $q<1/x$; the latter statement is true because the interval $[1,qx]$ is empty under this condition. Consequently, we have

[TABLE]

Remark 2.2**.**

The set of $k$-free integers defines a topological dynamical system $(\mathbb{X}_{k},\mathbb{Z})$, where $\mathbb{X}_{k}$ is the orbit closure of $V_{k}$ under the natural, continuous shift action of $\mathbb{Z}$ in the local topology. The frequency measure $\nu$ with respect to the chosen averaging sequence is well defined and known as the Mirsky measure. It is ergodic, and $V_{k}$ is a generic set for it; see [5] and references therein for details. The measure-theoretic dynamical system $(\mathbb{X}_{k},\mathbb{Z},\nu)$ has pure point dynamical spectrum, where the latter (in additive notation) is precisely the Abelian group $L^{\circledast}_{k}$ from above. This follows from [7] via the equivalence theorem on pure point spectra [6]; compare [5]. For the case $k=2$, it has also been derived by explicit means in [9]. Alternatively, it systematically follows from Keller’s new approach [13, 14]. $\Diamond$

To continue, we require the following preliminary results from [1, 12], where, for $n\in\mathbb{N}$, we use $d(n)$ to denote the number of positive divisors of $n$, $\sigma(n)$ for the sum of those divisors, and $\varphi(n)$ for Euler’s totient function, that is, $\varphi(n)=\varphi(1,n)$ with the function from (1.2).

Lemma 2.3**.**

For any integers $q,k\geqslant 2$ with $q\in\mathbb{N}_{k+1}$, we have

[TABLE]

Proof.

Under our conditions, the equality f_{k}(q)=\bigl{(}\varphi(\bar{q})\,\sigma(\bar{q}^{\hskip 0.5ptk-1})\bigr{)}^{-1} follows easily from the standard definitions of $\varphi$ and $\sigma$. Then, following the argument of [12, Thm. 329], we have

[TABLE]

Since $\varphi(\bar{q})=\bar{q}\prod_{p|\bar{q}}(1-p^{-1})$ and $\zeta(k)=\prod_{p}(1-p^{-k})^{-1}$, we clearly get

[TABLE]

which is the desired result. ∎

Lemma 2.4**.**

Let $x>0$ be fixed, $q\in\mathbb{N}_{k+1}$, and let $\ell=q/\bar{q}.$ Then, one has

[TABLE]

Moreover, for square-free $q$ — which means $q=\bar{q}\in\mathbb{N}_{2}$ — we have

[TABLE]

Consequently, we have the inequality

[TABLE]

Proof.

The first claim follows from the definition of $\varphi(x,q)$ in (1.2), since $\gcd(m,q)=1$ if and only if $\gcd(m,\bar{q})=1$, where $q=\ell\hskip 0.5pt\bar{q}$ with $\ell\hskip 0.5pt|\hskip 0.5pt\bar{q}^{k}$ under our assumption. The second and third relations follow from [10, Secs. 1 and 2]. ∎

Lemma 2.5**.**

Let $k\geqslant 2$ be an integer. For sufficiently small positive real $y$, one has

[TABLE]

Proof.

Let $M(t)\mathrel{\mathop{:}}=\sum_{q\leqslant t}\lvert\mu(q)\rvert$. By partial summation, we obtain

[TABLE]

Since M(t)=\frac{t}{\zeta(2)}+O\bigl{(}\sqrt{t}\,\bigr{)} for $t\in[1,\infty)$, see [12, Thm. 333], we find

[TABLE]

We now have all prerequisites to turn to our main result.

3. Proof of Theorem 1.1

We remind the reader that $k\geqslant 2$ is a fixed integer, thus any dependence on $k$ will be suppressed. Our arguments use a real parameter $\alpha\in[1,k]$, whose value will be specified later as a function of another parameter $\varepsilon$. In what follows, since we are ultimately interested in a limit as $x\to 0^{+}$, all big-$O$ terms are taken over $x\in(0,x_{0}(k)]$ for some sufficiently small $x_{0}(k)>0$. Note also that the constants in the big-$O$ symbols to follow do depend on $k$, but are independent of $\alpha$ in the range $\alpha\in[1,k]$.

Recall that a $k$-free $q$ has a unique decomposition $q=\ell\hskip 0.5pt\bar{q},$ where $\bar{q}$ is squarefree and $\ell\hskip 0.5pt|\hskip 0.5pt\bar{q}^{\hskip 0.5ptk-1}$, so that $q\leqslant\bar{q}^{\hskip 0.5ptk}$. By Eq. (2.1) and two applications of Lemma 2.4,

[TABLE]

Therefore, for $q^{\alpha}\geqslant\frac{1}{x_{0}(k)}$, we have, using Lemma 2.3 for the last two lines,

[TABLE]

where, as stated above, the implied constant in the big-$O$ term depends on the value of $k$, but is independent of $\alpha$ in the range $1\leqslant\alpha\leqslant k$.

Now let $\varepsilon\in(0,1)$ be arbitrary, but fixed. Using a result recorded in Hardy and Wright [12, Thm. 317], for $x$ small enough, and thus, correspondingly, for $q$ large enough, one has

[TABLE]

Consequently, we also have, for $x\leqslant x_{0}(k)$,

[TABLE]

Applying Lemma 2.5 with $y=x^{1/\alpha}$, and using only the leading term, we have

[TABLE]

where $O_{\varepsilon}$ is used to indicate the dependence of the implied constant on $\varepsilon$. Continuing the inequality for $Z_{k}(x)$ from above, and applying Lemma 2.5 yet again with $y=x^{1/\alpha}$, we obtain

[TABLE]

As before, the implied constants in the big-$O$ terms in Eq. (3.3) are independent of the parameter $\alpha\in[1,k]$. Therefore, we are free to choose $\alpha$ in that range, and in order to make the second remainder term asymptotically smaller than the first term on the right, we must have $\frac{k-1}{\alpha}+1<\frac{1}{\alpha}(2k-1-\varepsilon)$, which simplifies to

[TABLE]

Thus we let $\alpha=k(1-\varepsilon)$, so that $\alpha$ satisfies (3.4). Now, (3.3) becomes

[TABLE]

Since the final two big-$O$ terms have, for fixed $\varepsilon$, larger exponents than the main term, we obtain

[TABLE]

For an upper bound on the limit superior of \log\bigl{(}Z_{k}(x)\bigr{)}/{\log(x)}, as $x\to 0^{+}$, we employ the last assertion of Lemma 2.4, which states that $\varphi(x,q)\leqslant x\hskip 0.5pt\varphi(q)+d(q)$ for positive square-free integers $q$. As in the above argument, we let $\varepsilon\in(0,1)$ be arbitrary, but fixed, so that, for $x$ sufficiently close to zero, starting with Eq. (2.1) and applying Lemma 2.3, we get

[TABLE]

where for the last step, as above, we have used the inequality (3.2).

Applying Lemma 2.5, with $y=x^{1/k}$, to each of the sums in (3.6), for $x$ sufficiently close to zero, we have

[TABLE]

where, as before, we have written $O_{\varepsilon}$ to indicate the dependence of the implied constant on $\varepsilon$. Since the final two big-$O$ terms have, for fixed $\varepsilon$, larger exponents than the main term, we obtain

[TABLE]

Putting together (3.5) and (3.7) completes the proof of the theorem.

4. Further developments

The interested reader will note that the contributions from the Möbius and divisor sums can be made explicit using more delicate arguments from analytic number theory. In fact, with the use of such deep techniques, after seeing this paper on the arXiv, N. Rome and E. Sofos have proved a power-law asymptotic for $Z_{k}(x)$ using methods quite different from ours; see their preprint [19] for details.

Acknowledgements

It is our pleasure to thank Uwe Grimm for valuable discussions. We also thank the anonymous referee for a number of helpful suggestions, and N. Rome and E. Sofos for communicating with us about their result, which is now available on the arXiv (arXiv:1907.04845). Our work was supported by the Research Centre for Mathematical Modelling (RCM2) of Bielefeld University.