Finiteness and value of $M_4$, the supremum of reciprocal sums of sets of integers with no 4-term arithmetic progression

Let $M_k$ denote the supremum of the reciprocal sum $\sum_{n\in A}1/n$ over all sets $A$ of positive integers containing no nontrivial $k$-term arithmetic progression. Erdős's conjecture on arithmetic progressions (the large-sets conjecture, equivalent to $M_k<\infty$ for every $k$) was settled for $k=3$ by Bloom and Sisask and strengthened by Kelley and Meka, but for every $k\ge4$ even the finiteness of $M_k$ is unknown. For $k=4$ the best rigorous lower bound is $M_4\ge4.43975$, from digit-restricted constructions of Alexander Walker, and no upper bound of any size is known. The problem is to decide whether $M_4$ is finite -- equivalently, whether every 4-term-progression-free set of positive integers has a convergent reciprocal sum -- and, if finite, to determine or sharply bound its value.

Background

A set of positive integers is called large in the sense of Erdős if the sum of reciprocals of its elements diverges. Since any set of positive upper density (a set with $|A\cap\{1,\dots,N\}|\ge cN$ for some $c>0$ and infinitely many $N$) is automatically large, the following conjecture of Erdős on arithmetic progressions strengthens the classical density conjecture of Erdős and Turán (1936) that Szemerédi proved in 1975: every set of positive upper density contains arithmetic progressions of every finite length. Erdős's conjecture asserts that every large set contains arithmetic progressions of arbitrary finite length; the set of primes is large and indeed contains arbitrarily long progressions by the Green–Tao theorem. In the literature on reciprocal sums this conjecture is often called the Erdős–Turán conjecture on arithmetic progressions, following Gerver's naming.

Fix $k\ge3$ and let $\mathcal{S}_k$ be the family of sets of positive integers containing no nontrivial $k$-term arithmetic progression (a progression $x, x+y, \dots, x+(k-1)y$ with $y\ne0$; such sets are called $k$-free). Erdős's conjecture is equivalent to the finiteness, for every $k$, of

$M_k:=\sup_{A\in\mathcal{S}_k}\sum_{n\in A}\frac{1}{n},$

and Gerver proved in 1977 that the conjecture would imply a bound on each $M_k$ depending only on $k$ (a bound uniform in the set, as a function of $k$). The finitary shadow of the question is $r_k(N)$, the maximal size of a subset of $\{1,\dots,N\}$ containing no $k$-term progression: $M_k<\infty$ follows once $r_k(N)\ll N/(\log N)^{1+c}$ for some constant $c>0$, because the reciprocal sum over dyadic blocks is then summable.

The case $k=3$ is settled. Bloom and Sisask proved $r_3(N)\ll N/(\log N)^{1+c}$, breaking the logarithmic barrier in Roth's theorem (the $k=3$ density theorem of Roth from 1953 being the first case of Szemerédi's theorem), and this bounds the reciprocal sums of 3-free sets, giving $M_3<\infty$ -- the first nontrivial case of Erdős's conjecture. Kelley and Meka subsequently strengthened the density bound to quasipolynomial shape $2^{-O((\log N)^{1/12})}N$, with a further improvement by Bloom and Sisask to $\exp(-c(\log N)^{1/9})N$. All of these methods exploit the linear structure of 3-term progressions. For $k\ge4$ nothing remotely comparable is known: Szemerédi's theorem gives only $r_4(N)=o(N)$; the best bound for 4-term progressions, due to Green and Tao, is $r_4(N)\ll N/(\log N)^{c}$ for some absolute constant $c>0$, a genuine logarithmic saving but with an exponent far below the $1+c$ that convergence of reciprocal sums would require; and for $k\ge5$ the best bounds (even after the recent 2024 improvement of Leng, Sah, and Sawhney via a quasipolynomial inverse theorem for the Gowers uniformity norms -- the norms that measure the pseudorandomness relevant to longer progressions) save only factors of $\log\log$ scale. Consequently, whether $M_4$ is finite is open, and no explicit upper bound for $M_4$, finite or otherwise, is known.

Lower bounds for $M_k$ come from explicit constructions. Berlekamp proved $M_k\ge\frac{1}{2}k\log2$, and Gerver improved this to $M_k>(1-o(1))k\log k$. For small $k$ the benchmarks are numerical. Gerver and Ramsey studied the greedy sets $G_k$ (the lexicographically earliest $k$-free sets) and estimated heuristically that the reciprocal sum of $G_4$ is about $4.3$; Wróblewski proved the record $M_3\ge3.00849$ by interlacing a greedy set with a denser finite construction. Alexander Walker advanced the lower-bound theory using digit-restricted sets: for a base $b\ge2$ and a proper digit set $S\subsetneq[0,b-1]$, the Kempner set $\mathcal{K}(S,b)$ is the set of non-negative integers whose base-$b$ digits all lie in $S$. If $S$ contains no $k$-term arithmetic progression modulo $b$ with common difference not divisible by $b$, and $0\in S$, then $\mathcal{K}(S,b)$ is $k$-free in the integers, and the reciprocal sum of the shifted set $\mathcal{K}(S,b)+1$ can be computed to arbitrary precision with rigorous error control. Walker proved that shifted Kempner sets approximate the reciprocal sum of any $k$-free set with convergent reciprocal sum arbitrarily well, and a large branch-and-bound search over digit sets yielded $M_4\ge4.43975$ (a 21-element digit set in base 55) and $M_{10}\ge14.056$, with no improvement found for $M_4$ in bases up to 200 under heuristic pruning.

Problem Statement

Decide whether

$M_4=\sup\left\{\sum_{n\in A}\frac{1}{n}:\ A\subseteq\mathbb{Z}_{>0}\ \text{contains no nontrivial 4-term arithmetic progression}\right\}$

is finite. Here a nontrivial 4-term arithmetic progression means a progression $x, x+d, x+2d, x+3d$ with $x,d$ integers and $d\ne0$. Equivalently: must every set of positive integers whose reciprocal sum diverges contain a nontrivial 4-term arithmetic progression? This is the $k=4$ case of Erdős's conjecture on large sets, and it is the first case not covered by known results: the $k=3$ case follows from Bloom and Sisask's bound $r_3(N)\ll N/(\log N)^{1+c}$, while for $k=4$ the best known density bound $r_4(N)\ll N/(\log N)^{c}$ (Green and Tao) has an exponent far below what convergence of reciprocal sums requires.

If $M_4$ is finite, determine its value, or establish explicit two-sided bounds for it; at present the only rigorous information is $M_4\ge4.43975$ (Walker's shifted digit-restricted construction in base 55), and no upper bound of any size is known. More generally, the same finiteness question stands for every $k\ge4$, and finiteness of $M_k$ for all $k$ is equivalent to the full conjecture; $k=4$ is its first open instance.

References

1. Alexander Walker, Integer Sets of Large Harmonic Sum which Avoid Long Arithmetic Progressions, arXiv:2203.06045 (2022; v2 September 2025), doi:10.48550/arXiv.2203.06045, https://arxiv.org/abs/2203.06045
2. Thomas F. Bloom and Olof Sisask, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions, arXiv:2007.03528 (2020), doi:10.48550/arXiv.2007.03528, https://arxiv.org/abs/2007.03528
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