Erdős's reciprocal-sum conjecture for arithmetic progressions: the open cases $k \ge 4$

Erdős conjectured, repeatedly from the 1940s onward, that any set $A$ of natural numbers whose reciprocal sum diverges, $\sum_{n\in A} 1/n=\infty$, must contain infinitely many arithmetic progressions of every length $k\ge 3$. The first nontrivial case $k=3$ was settled affirmatively by Bloom and Sisask (2020) through the bound $r_3(N)\ll N/(\log N)^{1+c}$, and Pilatte's paper on Roth's theorem with generalized coefficients (Discrete Analysis, 2022) explicitly records that the general case $k\ge 4$ seems to be well beyond the reach of current techniques. The general case $k\ge 4$ remains open: no resolution, and no quantitative progress of the requisite strength on these cases, is known as of 2026. The first open instance is whether every set of natural numbers with divergent reciprocal sum must contain a 4-term arithmetic progression; the corresponding quantitative question is whether $r_4(N)\ll N/(\log N)^{1+c}$ holds for some absolute $c>0$, a bound which would imply the $k=4$ case by partial summation.

Background

An arithmetic progression of length $k$ (a $k$-term AP) is a sequence $a, a+d, \dots, a+(k-1)d$ with $d\ne 0$. Szemerédi's theorem (1975) states that any subset of the natural numbers with positive upper density (meaning that the limit superior of $|A\cap\{1,\dots,N\}|/N$ as $N\to\infty$ is positive) contains $k$-term APs of every length, and its quantitative finite form is phrased through $r_k(N)$, the largest size of a subset of $\{1,\dots,N\}$ containing no nontrivial $k$-term AP; Szemerédi's theorem is the statement $r_k(N)=o(N)$. The case $k=3$ is Roth's theorem (1953), whose proof introduced the Fourier-analytic density-increment method — locating a structured sub-piece of the ambient set on which the density increases, and iterating — which underlies most subsequent work.

Erdős's conjecture concerns a different, logarithmic notion of size: a set can have zero upper density (so Szemerédi's theorem does not apply) yet have a divergent sum of reciprocals. The conjecture predicts that any $A\subseteq\mathbb{N}$ with $\sum_{n\in A}1/n=\infty$ contains infinitely many $k$-term APs for every $k\ge 3$. The motivating example is the primes: Euler's theorem that $\sum_p 1/p$ diverges, and the Green–Tao theorem that the primes contain arbitrarily long APs, suggest that density in the reciprocal-sum sense alone should force progressions. By partial summation (the standard calculus identity converting information about a sum over a set into information about its growth), the conjecture at a fixed length $k$ would follow from a bound of the shape $r_k(N)\ll N/(\log N)^{1+c}$ for some absolute constant $c>0$.

For $k=3$ this program is complete: after decades of partial results, Bloom and Sisask (2020) proved $r_3(N)\ll N/(\log N)^{1+c}$, settling the $k=3$ case of Erdős's conjecture, and Kelley and Meka (2023), as expounded by Bloom and Sisask, subsequently improved this to $r_3(N)\le\exp(-c(\log N)^{1/12})N$. For $k\ge 4$ the situation is radically different: the best known upper bounds remain of log-log quality, namely Gowers's bound $r_4(N)\ll N/(\log\log N)^{c}$ from his new proof of Szemerédi's theorem via higher-order Fourier analysis (an extension of classical Fourier analysis that decomposes functions against polynomial rather than linear phase functions), and the bounds $r_k(N)\ll N/\exp((\log\log N)^{c})$ for $k\ge 5$ of Leng, Sah, and Sawhney (2023 for $k=5$, extended to all $k\ge 5$ in 2024) — all far weaker than the logarithmic strength that partial summation requires.

The source recording this status is Pilatte's paper (published in Discrete Analysis, 2022), which proves a conjecture of Shkredov and Solymosi: every $A\subseteq\mathbb{Z}^2$ with $\sum_{a\in A\setminus\{0\}}1/\|a\|^{2}=\infty$ contains infinitely many isosceles right triangles, obtained by adapting the Bloom–Sisask machinery to linear equations $T_1a_1+T_2a_2+T_3a_3=0$ in finite abelian groups whose coefficients $T_i$ are group automorphisms. Its introduction records the status of Erdős's conjecture — $k=3$ resolved, the general case beyond current techniques — and its Remark 1.7 records, as a companion open problem, Graham's conjecture that such a divergent-sum subset of the square lattice must contain infinitely many axis-parallel squares, believed comparable in difficulty to the $k=4$ case of Erdős's conjecture. Recent work on grid analogues (square-free and triangle-free subsets of the $n\times n$ grid) confirms these divergent-sum questions remain active, with large gaps between constructions and bounds, but leaves both the $k\ge 4$ Erdős cases and Graham's conjecture untouched.

Problem Statement

Erdős's conjecture on arithmetic progressions (posed repeatedly from the 1940s; recorded in Pilatte's paper, Discrete Analysis, 2022): if $A\subseteq\mathbb{N}$ satisfies $\sum_{n\in A}1/n=\infty$, must $A$ contain infinitely many $k$-term arithmetic progressions for every $k\ge 3$? The case $k=3$ is affirmatively resolved, so the open content of the conjecture is the cases $k\ge 4$, of which the first is: must every $A\subseteq\mathbb{N}$ with $\sum_{n\in A}1/n=\infty$ contain a 4-term arithmetic progression? (One progression implies infinitely many: after deleting the finitely many elements of the progressions found so far, the reciprocal sum still diverges, so another progression must exist.) The first open case also has the quantitative form: does $r_4(N)\ll N/(\log N)^{1+c}$ hold for some absolute constant $c>0$, where $r_4(N)$ is the largest size of a subset of $\{1,\dots,N\}$ with no nontrivial 4-term arithmetic progression? The divergent-sum statement for $k=4$ then follows by partial summation. The conjecture is settled positively only by a proof covering every length $k\ge 4$; a proof at a single length settles that case but leaves the general conjecture open, while a set $A\subseteq\mathbb{N}$ with $\sum_{n\in A}1/n=\infty$ containing no $k$-term arithmetic progression for some length $k\ge 4$ would refute it.

References

1. Cédric Pilatte, New bound for Roth's theorem with generalized coefficients, Discrete Analysis (2022), 2022:16, 21 pp. Journal DOI: 10.19086/da.55553 (displayed by the journal and by arXiv, but not registered with Crossref or doi.org); arXiv:2111.03838. Publisher page: https://discreteanalysisjournal.com/article/55553 ; https://arxiv.org/abs/2111.03838
2. Thomas F. Bloom and Olof Sisask, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions, arXiv:2007.03528 (2020, v2 2021). https://arxiv.org/abs/2007.03528
3. Sarah Peluse, Recent progress on bounds for sets with no three terms in arithmetic progression, Séminaire Bourbaki, exposé 1196, Astérisque 438 (2022), 547–581. DOI: 10.24033/ast.1197; arXiv:2206.10037. https://arxiv.org/abs/2206.10037
4. Zander Kelley and Raghu Meka, Strong bounds for 3-progressions, in 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), 2023. DOI: 10.1109/focs57990.2023.00059; arXiv:2302.05537. https://arxiv.org/abs/2302.05537
5. Thomas F. Bloom and Olof Sisask, The Kelley–Meka bounds for sets free of three-term arithmetic progressions, Essential Number Theory 2 (2023), no. 1, 15–44. DOI: 10.2140/ent.2023.2.15; arXiv:2302.07211. https://arxiv.org/abs/2302.07211
6. James Leng, Ashwin Sah, and Mehtaab Sawhney, Improved bounds for five-term arithmetic progressions, arXiv:2312.10776 (2023). https://arxiv.org/abs/2312.10776
7. James Leng, Ashwin Sah, and Mehtaab Sawhney, Improved bounds for Szemerédi's theorem, arXiv:2402.17995 (2024). https://arxiv.org/abs/2402.17995
8. W. T. Gowers, A new proof of Szemerédi's theorem, Geometric and Functional Analysis 11 (2001), 465–588. DOI: 10.1007/s00039-001-0332-9. https://doi.org/10.1007/s00039-001-0332-9
9. Gyula Károlyi and Jozsef Solymosi, A Salem–Spencer-type construction for large subsets of integer grids with no isosceles right triangles, arXiv:2607.22828 (2026). https://arxiv.org/abs/2607.22828
10. Máté Jánosik, Artúr Nádor, Zoltán Lóránt Nagy, and László Bence Simon, Avoiding configurations of small size in the square grid, arXiv:2601.14465 (2026). https://arxiv.org/abs/2601.14465
11. Ilya Shkredov and Jozsef Solymosi, Tilted corners in integer grids, in Number Theory and Combinatorics, De Gruyter (2022), 329–338. DOI: 10.1515/9783110754216-020; arXiv:2103.01645. https://arxiv.org/abs/2103.01645