Approximate arithmetic structure in large sets of integers

TL;DR

This paper demonstrates that large sets of integers, as defined by Erdős, can closely approximate arbitrarily long arithmetic progressions with a quantifiable error bound, improving previous results.

Contribution

It establishes a stronger quantitative approximation of long arithmetic progressions within large sets of integers, refining earlier bounds from o(Δ) to O(Δ^α) for any α in (0,1).

Findings

Improved approximation bounds for arithmetic progressions in large sets.

Quantitative error estimates expressed as O(Δ^α).

Extension of previous results from o(Δ) to O(Δ^α).

Abstract

We prove that if a set is `large' in the sense of Erd\H{o}s, then it approximates arbitrarily long arithmetic progressions in a strong quantitative sense. More specifically, expressing the error in the approximation in terms of the gap length $\Delta$ of the progression, we improve a previous result of $o(\Delta)$ to $O(\Delta^\alpha)$ for any $\alpha \in (0,1)$.

Equations

$$\underset{p\in P}{\sup }\underset{a\in A}{\inf }|p-a|\le \epsilon \Delta .$$

$$\#A\cap [0,n]\ge \frac{n}{(\log n{)}^{\gamma }}$$

$$\underset{p\in P}{\sup }\underset{a\in A}{\inf }|p-a|\le {\Delta }^{\alpha }.$$

$$\#A\cap [0,n]\ge \frac{n}{(\log n{)}^{\gamma }}$$

$$\sum _{a\in A}{a}^{-1}\le \sum _{a\in A\cap [1,e^t]}\frac{1}{a}+\sum _{n\ge t}\sum _{a\in A\cap [e^n,e^{n+1}]}{e}^{-n}\le 2t+e\sum _{n\ge 1}{n}^{-\gamma }<\infty .$$

$$\underset{p\in P}{\sup }\underset{a\in A_n}{\inf }|p-a|\le 2{\Delta }^{\alpha },$$

$$(2\epsilon {)}^2\frac{N}{N^{{\alpha }^2}}\frac{N^{{\alpha }^2}}{N^{{\alpha }^4}}$$

$$\frac{1}{2}{N}^{{\alpha }^{2m^{\prime }}}/N^{{\alpha }^{2m^{\prime }+1}}>M_{\epsilon },$$

$$m^{\prime }\approx \frac{\log \log N+\log (1-\alpha )-\log \log M_{\epsilon }}{-2\log \alpha }.$$

$$\Delta \ge N^{{\alpha }^{2m^{\prime }+2}}\lfloor N^{{\alpha }^{2m^{\prime }+1}}/N^{{\alpha }^{2m^{\prime }+2}}\rfloor \ge \frac{1}{2}{N}^{{\alpha }^{2m^{\prime }+1}}\approx 1$$

$$m=m^{\prime }-\frac{\log \log \log N}{-2\log \alpha }.$$

$$\Delta \ge \frac{1}{2}{N}^{{\alpha }^{2m+1}}\gtrsim n,$$

$$(2\epsilon {)}^{m+1}\frac{N}{N^{{\alpha }^2}}\frac{N^{{\alpha }^2}}{N^{{\alpha }^4}}\cdots \frac{N^{{\alpha }^{2m}}}{N^{{\alpha }^{2m+2}}}\left(2N^{{\alpha }^{2m+2}}\right)=2(2\epsilon {)}^{m+1}N$$

$$\#A_n\le 2(2\epsilon {)}^{m+1}N\lesssim (2\epsilon {)}^m{2}^n\lesssim {\left(\frac{\log n}{n}\right)}^{\frac{\log 2\epsilon }{2\log \alpha }}{2}^n$$

$${\left(\frac{\log n}{n}\right)}^{\frac{\log 2\epsilon }{2\log \alpha }}\le n^{-\gamma }$$

$$\#A\cap [0,T]\le \sum _{n=0}^{\lceil \frac{\log T}{\log 2}\rceil }\#A_n\lesssim \sum _{n=0}^{\lceil \frac{\log T}{\log 2}\rceil }{n}^{-\gamma }{2}^n\lesssim \frac{T}{(\log T{)}^{\gamma }}.$$

$$\underset{p\in P}{\sup }\underset{a\in A_t}{\inf }|p-a|>{\Delta }^{\alpha }.$$

$$r^t+s^t=2.$$

$$\{(x,y)\in {\mathbb{R}}^2:x^t+y^t=2\},$$

$$f_t:=\underset{b>a\ge 10}{\inf }\frac{\log {\min }_{n\in \mathbb{N}}|\frac{a^t+b^t}{2}-n^t|}{\log |b^t-a^t|}?$$

$$f_t(b)=\underset{b>a\ge 10}{\min }\frac{\log {\min }_{n\in \mathbb{N}}|\frac{a^t+b^t}{2}-n^t|}{\log |b^t-a^t|}$$

$$x^3+y^3-2z^3\in \{\pm 1,\pm 2\}.$$

$$\#A\cap \{x\in {\mathbb{Z}}^d:|x|\in [0,n]\}\ge \frac{n^d}{(\log n{)}^{\gamma }}$$

$$\underset{p\in \Delta C+t}{\sup }\underset{a\in A}{\inf }|p-a|\le {\Delta }^{\alpha },$$

$$r_k(N)\le C_k\frac{N}{(\log \log N{)}^{c_k}}$$

$$\#A\cap [0,n]\ge \frac{n}{(\log n{)}^{\gamma }}$$

Full text

1. The Erdős conjecture on arithmetic progressions

The Erdős conjecture on arithmetic progressions is a famous open problem in combinatorial number theory, which states that a set of positive integers whose reciprocals form a divergent series111 Erdős used the term ‘large’ to describe such sets. should contain arbitrarily long arithmetic progressions. More precisely, if $A\subset\mathbb{N}$ is such that $\sum_{a\in A}a^{-1}=\infty$, then $A$ should contain an arithmetic progression of length $k$ for all $k\geq 1$. Note that the Green-Tao theorem, where $A$ is the set of prime numbers, is a special case of this conjecture, see [GT08].

In [FY18], we established the following weak version of this conjecture. We say an arithmetic progression $\{a,a+\Delta,\dots,a+(k-1)\Delta\}\subset\mathbb{R}$ is of length $k\in\mathbb{N}$ and gap length $\Delta>0$.

Theorem 1.1** (Theorem 2.1 in [FY18]).**

If $A\subset\mathbb{N}$ is such that $\sum_{a\in A}a^{-1}=\infty$, then for all $\varepsilon>0$ and all $k\geq 1$, there exists an arithmetic progression $P$ of length $k$ and gap length $\Delta\geq 1$ such that

[TABLE]

This result should be interpreted as saying that $A$ gets arbitrarily close to arbitrarily long arithmetic progressions, see also [Fr19]. One disadvantage of this result is that the conclusion holds for natural examples of sets which are known not to contain long arithmetic progressions, such as the squares. Moreover, there is no information on how $P$ depends on $\varepsilon$ and $k$. Another direction in which this result could be improved is to allow $\varepsilon$ to depend on $\Delta$, that is to improve the bound on the ‘uncertainty’ from $o(\Delta)$ to something stronger. For example, if one could show that $\varepsilon$ could be replaced by $C\Delta^{-1}$ for some constant $C,$ then the Erdős conjecture would follow. This is because our ‘uncertainty’ $\varepsilon\Delta$ is at most $C$ and the result would then follow from Van der Waerden’s Theorem, see [vdW27]. More precisely, let $P$ be a very long arithmetic progression which our set approximates to within $C$. Colour the points in our set which approximate $P$ according to their distance from $P$ (in particular we need at most 2$C$+1 colours). Van der Waerden’s Theorem allows us to extract a monochromatic arithmetic progression of length $k$ provided the length of $P$ is large enough. Since this progression is monochromatic it is a genuine arithmetic progression of length $k$ inside our set.

The main result of this paper addresses each of the above issues. For example, the conclusion of our main result should not hold for sets such as the squares or cubes (although proving this rigorously seems challenging and is related to Mazur’s near miss problem, see Section 4), and we get quantitative information regarding $P$. We essentially show that one can choose $\varepsilon=C\Delta^{-\delta}$ for any $0<\delta<1.$ Thus, we push Theorem 1.1 further towards Erdős conjecture, which would follow if we could choose $\delta=1$. The ideas in this paper can be used to push the result even further, relying on a result of Gowers [G01], see Section 5.

2. Main result

Our main result is the following improvement over Theorem 1.1. We write $\#E$ to denote the cardinality of a finite set $E$.

Theorem 2.1**.**

Suppose $A\subset\mathbb{N}$ is such that there exists a constant $\gamma>0$ such that

[TABLE]

for infinitely many $n$. Then, for all $\alpha\in(0,1),k\geq 1,\Delta_{0}>1$, there exists infinitely many arithmetic progressions $P$ of length $k$ and gap length $\Delta\geq\Delta_{0}$ such that

[TABLE]

Moreover, there is a constant $c>0$, depending only on $\alpha,\gamma$, such that for infinitely many $n\in\mathbb{N}$, $P$ can be chosen to have gap length at least $cn$ and lie in the interval \big{[}2^{n},\,2^{n+1}\big{]}.

Since sets of integers whose reciprocals form a divergent series necessarily satisfy the power-log density assumption above (with $\gamma>1$), Theorem 2.1 applies to sets which are ‘large’ in the sense of Erdős. More precisely, if $A\subset\mathbb{N}$ is such that $\sum_{a\in A}a^{-1}=\infty$, then for all $\gamma>1$, we have

[TABLE]

for infinitely many $n$. Indeed, if (*) were not true for some $\gamma>1$ and all $n\geq e^{t}$ for some positive integer $t$, then

[TABLE]

Finally, we note that Theorem 1.1 follows directly from Theorem 2.1 as follows. Fix $\varepsilon>0$ and $k\geq 1$ and apply Theorem 2.1 with $\alpha=1/2$ and $\Delta$ chosen large enough to ensure that $\sqrt{\Delta}\leq\varepsilon\Delta$.

3. Proof of Theorem 2.1

Fix $\alpha\in(0,1)$, $k\geq 3$ and $\gamma>0$. Let $\varepsilon\in(0,1/2)$ and $M_{\varepsilon}$ be an integer such that, for all $M>M_{\varepsilon}$, any subset of $\{1,\dots,M\}$ with at least $\varepsilon M$ elements must contain an arithmetic progression of length $k$. Such a number $M_{\varepsilon}$ exists as a direct consequence of Szemerédi’s celebrated theorem on arithmetic progressions, [S75]. Later we will fix a particular $\varepsilon$ depending on $\alpha$ and $\gamma$.

We write $X\lesssim Y$ to mean that $X\leq cY$ for some universal constant $c>0$. We also write $X\gtrsim Y$ to mean $Y\lesssim X$ and $X\approx Y$ to mean that both $X\lesssim Y$ and $X\gtrsim Y$ hold. The implicit constants $c$ appearing here can, and often will, depend on $\alpha,k,\gamma,\varepsilon$. We write $\lfloor x\rfloor$ to denote the integer part of $x\geq 0$.

Consider the dyadic intervals $[2^{n},2^{n+1})$ for integers $n\geq 0$ and let $A_{n}=A\cap[2^{n},2^{n+1}).$ For notational simplicity we write $N=2^{n}.$ Decompose $[2^{n},2^{n+1})$ into smaller (half-open) intervals of equal length $N^{\alpha^{2}}$ and label these intervals from left to right by $1,2,\dots$. We may not be able to perform this decomposition exactly, in which case we will be left with an interval of length $N-\lfloor N/N^{\alpha^{2}}\rfloor N^{\alpha^{2}}$ at the right hand side, which we simply discard. Group these intervals into equivalence classes by considering their labels modulo $\lfloor N^{\alpha}/N^{\alpha^{2}}\rfloor$. Note that the set of centres of intervals in a given equivalence class form an arithmetic progression of length (at least) $\tfrac{1}{2}N/N^{\alpha}$ and gap length $N^{\alpha^{2}}\lfloor N^{\alpha}/N^{\alpha^{2}}\rfloor$.

Suppose that $\tfrac{1}{2}N/N^{\alpha}>M_{\varepsilon}$, which is certainly true for all sufficiently large $n$. In order to avoid the existence of an arithmetic progression $P$ with length $k$ and gap length $\Delta=N^{\alpha^{2}}\lfloor N^{\alpha}/N^{\alpha^{2}}\rfloor$ such that

[TABLE]

we see that $A_{n}$ can intersect no more than $2\varepsilon N/N^{\alpha^{2}}$ many intervals of length $N^{\alpha^{2}}.$ Indeed, each equivalence class contains at least $M_{\varepsilon}$ and at most $2N/N^{\alpha}$ many intervals and so must intersect $A_{n}$ in fewer than $2\varepsilon N/N^{\alpha}$ many of these intervals, and there are fewer than $N^{\alpha}/N^{\alpha^{2}}$ many equivalence classes.

We repeat the above argument inside each interval of length $N^{\alpha^{2}}$ that intersects $A_{n}.$ That is, we decompose each interval of length $N^{\alpha^{2}}$ into intervals of equal length $N^{\alpha^{4}}$ (possibly discarding a small piece at the end) and then we work modulo modulo $\lfloor N^{\alpha^{3}}/N^{\alpha^{4}}\rfloor$. In particular, if $\tfrac{1}{2}N^{\alpha^{2}}/N^{\alpha^{3}}>M_{\varepsilon}$ and there does not exist an arithmetic progression $P$ with length $k$ and gap length $\Delta=N^{\alpha^{4}}\lfloor N^{\alpha^{3}}/N^{\alpha^{4}}\rfloor$ satisfying (1), then $A_{n}$ can intersect no more than

[TABLE]

many intervals of length $N^{\alpha^{4}}.$ We can repeat this decomposition argument $(m^{\prime}+1)$ times where $m^{\prime}$ is chosen to be the largest integer satisfying

[TABLE]

noting that

[TABLE]

However, repeating the argument this many times means we can only bound the gap length of the arithmetic progressions we avoid below by

[TABLE]

which is not sufficient to prove the theorem. Therefore we choose to repeat the argument only $(m+1)$ times where

[TABLE]

At the $(l+1)$st step we decompose intervals of length $N^{\alpha^{2l}}$ into intervals of length $N^{\alpha^{2l+2}}$ and work modulo $\lfloor N^{\alpha^{2l+1}}/N^{\alpha^{2l+2}}\rfloor$.

After applying this argument $(m+1)$ times we see that if there does not exist an arithmetic progression $P$ satisfying (1) with gap length

[TABLE]

then $A_{n}$ contains at most

[TABLE]

many elements, where we used the fact that intervals of length $N^{\alpha^{2m+2}}$ contain at most $N^{\alpha^{2m+2}}+1\leq 2N^{\alpha^{2m+2}}$ many integers. In particular,

[TABLE]

where the implicit constants here depend on $\alpha$ and $\varepsilon$.

In order to reach a contradiction, suppose that for all but finitely many $n$ there does not exist an arithmetic progression $P$ satisfying (1). Therefore, the above cardinality estimate for $A_{n}$ holds for all but finitely many $n$. We now fix $\varepsilon>0$ depending on $\alpha$ and $\gamma$ such that

[TABLE]

for all $n\geq 1$. Therefore, for integer $T>0$,

[TABLE]

This contradicts the power-log density hypothesis since $\gamma>0$ can be chosen to be arbitrarily large. Therefore, for infinitely many $n$, there exists an arithmetic progression $P\subset[2^{n},2^{n+1})$ satisfying (1) with gap length $\Delta\gtrsim n$, proving the theorem. Note that the upper bound of $2\Delta^{\alpha}$ in (1) can be trivially upgraded to the desired upper bound of $\Delta^{\alpha}$ by replacing $\alpha$ with $\alpha^{\prime}\in(0,\alpha)$ in the argument and choosing $\Delta$ large enough.

4. A remark on Mazur’s near miss problem

From Theorem 2.1 we see that all ‘large enough’ sets $A\subseteq\mathbb{N}$ must ‘nearly’ contain arbitrarily long arithmetic progressions. It is interesting to consider examples of sets $A$ for which the conclusion of Theorem 1.1 is satisfied but the conclusion of Theorem 2.1 fails. It is easy to show existence of such examples but we are particularly interested in the sets $A_{t}=\{n^{-t}:n\in\mathbb{N}\}$ for a fixed integer $t\geq 2$, which turn out to be elusive and related to a deep problem posed by Mazur. It follows from [FY18] that $A_{t}$ satisfies the conclusion of Theorem 1.1 since the set of reciprocals of elements in $A_{t}$ is a set of full Assouad dimension. However, it is known that these sets do not contain genuine arithmetic progressions of length $k\geq 4$ (or $k\geq 3$ provided $t\geq 3$). We make the following conjecture.

Conjecture 4.1**.**

Let $A_{t}=\{n^{-t}:n\in\mathbb{N}\}$ for a fixed integer $t\geq 2$ and let $\alpha\in(0,1)$. There exist integers $k_{0}\geq 3$ and $\Delta_{0}\geq 1$ such that if $P$ is an arithmetic progression of length $k\geq k_{0}$ and gap length $\Delta\geq\Delta_{0}$, then

[TABLE]

The above conjecture is related to Mazur’s near miss problem, see [M04, Section 11]. To illustrate the connection, let us only consider arithmetic progressions of length 3. If $a,b,c$ forms an arithmetic progression then $a+c=2b.$ Suppose that $a,b,c$ are $t$-th powers, in which case we can find rational numbers $r$ and $s$ such that

[TABLE]

Our goal is not finding exact progressions in the set of $t$-th powers - indeed, there are no arithmetic progressions of length 3 inside the cubes. Instead, we want to know how close the set of $t$-th powers can get to an arithmetic progression of length 3. Given a large positive integer $Q$, we are interested in estimating the smallest distance between points on the lattice $\mathbb{Z}^{2}/Q$ and the curve defined by

[TABLE]

which is in the spirit of Mazur’s near miss problem. More specifically, Conjecture 4.1 is related to bounding this distance from below by $Q^{-\alpha}$ for $\alpha\in(0,1).$ From here, it is natural to consider the following question.

Question 4.2**.**

Given an integer $t\geq 3$, what is

[TABLE]

In order to test this question numerically, we computed the values

[TABLE]

for integer values of $b$ up to $10,000$. The results are plotted in Figure 1. This simulation suggests that $f_{3}=0$ but that $f_{t}>0$ for $t\geq 4$, and we believe this is the case.

Conjecture 4.3**.**

For integers $t\geq 4$, $f_{t}>0.$ On the other hand $f_{3}=0$ and there are infinitely many integer solutions to

[TABLE]

5. Further comments

5.1. Higher dimensions

Szemerédi’s theorem can be generalized for studying ‘structures’ in large subsets of $\mathbb{Z}^{d},d\geq 2,$ see [FK78]. This can help us get a higher dimensional version of Theorem 2.1 with an almost identical proof as in Section 3.

Theorem 5.1**.**

Let $C\subset\mathbb{Z}^{d}$ be a finite set222We use $C$ for Constellation as suggested in [TZ15].. Suppose $A\subset\mathbb{Z}^{d}$ is such that there exists a constant $\gamma>0$ such that

[TABLE]

for infinitely many $n$. Then, for all $\alpha\in(0,1),k\geq 1,\Delta_{0}>1$, there exists infinitely many translates $t\in\mathbb{Z}^{d}$ and $\Delta\geq\Delta_{0}$ such that

[TABLE]

where $\Delta C+t$ is the image of $C$ under the similitude $z\mapsto\Delta z+t$.

5.2. Reducing the uncertainty

In proving Theorem 2.1, we used Szemerédi’s theorem. In fact, one can obtain a slightly better result by performing a more careful analysis. Let $N_{1},N_{2}\geq 2$ be integers. Divide $[0,1]$ into $N_{1}$ equal pieces and then divide each of these small pieces into $N_{2}$ pieces of equal length. In total, we have $N_{1}N_{2}$ small pieces of length $(N_{1}N_{2})^{-1}.$ These pieces can be grouped into $N_{2}$ many ‘arithmetic progressions’ (more precisely speaking, intervals whose centres form an arithmetic progression) of length $N_{1}$ with gap $N_{1}^{-1}.$ To proceed further, we use the standard notation $r_{k}(N)$ to denote the largest cardinality of a subset of $\{1,\dots,N\}$ which contains no arithmetic progressions of length $k$. We want to select a certain number of small pieces of length $(N_{1}N_{2})^{-1}$ such that we do not get any arithmetic progressions of length of length $k$ with gap $N_{1}^{-1}.$ This number can be bounded above by $N_{2}r_{k}(N_{1}).$ For each $\varepsilon>0,$ if $N_{1}$ is large enough then we can replace $r_{k}(N_{1})$ by $\varepsilon N_{1}.$ This is what we did in the proof of Theorem 2.1. However, there are now stronger quantitative upper bounds for $r_{k}(N)$ than the Szemerédi bound of $\varepsilon N$. For example, we have the following result due to Gowers, see [G01].

Theorem 5.2**.**

(Gowers) For each $k\geq 3,$ there are constants $c_{k},C_{k}>0$ such that

[TABLE]

for all $N\geq 100.$ Here $c_{k}$ can be chosen as $2^{-2^{k+9}}.$

By applying this result, it should be possible to improve Theorem 2.1 by replacing $\Delta^{\alpha}$ with $f(\Delta)$ for a suitable increasing function $f$. One can in principle compute $f$ precisely but we decided not to pursue the details. We suspect that $f(\Delta)=\Delta^{1/\log\log\log\Delta}$ will probably do the job. (This is obtained by replacing $\alpha$ in $(**)$ with $1/\log\log\log N$.)

Finally, we remark that the arguments in this paper provide a road map for translating estimates for $r_{k}(N)$ into statements of the type presented in Theorem 2.1. In fact, if one could establish sufficiently good estimates for $r_{k}(N)$, then one could prove the Erdős conjecture.

5.3. Sharpness and non-integer sets

Instead of trying to reduce the uncertainty, $f(\Delta)$, as discussed in the previous subsection, consider $f(\Delta)=\Delta^{\alpha}$ for some $\alpha\in(0,1)$, as in Theorem 2.1. What is interesting now is improving the ‘largeness’ condition

[TABLE]

in the statement of Theorem 2.1. In general, one can try to prove Theorem 2.1 with $n/(\log n)^{\gamma}$ being replaced by a general increasing function $g(n).$ A natural question to ask is whether $g(n)$ can be chosen to be $n^{\delta}$ for a $\delta>0.$ As we have seen in Section 4, this is probably not true for $\delta=1/2.$ Also we suspect that $\delta$ can depend on $\alpha.$

Our argument works not only for integer sets. Indeed, if we consider $A\subset\mathbb{R}^{+}$ and require that $A$ is uniformly $\delta$-discrete for a number $\delta>0$, i.e. $\inf_{a,b\in A}|a-b|>\delta>0,$ then all the arguments in the proof of Theorem 2.1 apply in this case and we have the same result. Clearly, in this non-integer case, one cannot hope to find exact arithmetic progressions since small perturbations can destroy all exact progressions. Thus, the notion of ‘almost arithmetic progressions’ we introduce here is very natural. In this case, one can try to lowering the ‘uncertainty’ $f(\Delta)$ which was discussed in the previous paragraph.

6. Acknowledgements

JMF acknowledges financial support from an EPSRC Standard Grant (EP/R015104/1). HY was financially supported by the University of St Andrews. The authors thank Sam Chow for suggesting Mazur’s near miss problem.