Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds

1 · The essentials

An arithmetic progression is a list of numbers with the same positive gap between consecutive entries, such as 2, 5, 8. The manuscript studies when a collection of positive whole numbers must contain such lists of any desired finite length. Its main result says that this happens whenever adding the reciprocals of the numbers in the collection gives a sum that grows without bound. This is the statement identified in the source as Erdős’s conjecture. The manuscript also gives a quantitative limit on how many numbers can be chosen from 1 through N while avoiding a list of a fixed length. The allowed fraction becomes smaller as N grows, at a rate specified by the result. The main result has a Lean formalization, although the supplied material does not specify its exact coverage.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine numbered lockers, with each selected locker numbered n contributing a token worth 1/n. The result says that if the accumulated value never stops growing as more lockers are considered, the selected locker numbers must include equally spaced lists of every finite length. The image illustrates the condition, but does not explain why it forces those patterns.

Why it matters

The result connects a condition on an infinite sum with the presence of exact patterns among integers. Its quantitative statement also controls finite collections: for each fixed progression length, it limits the size of a collection that avoids that pattern. The manuscript presents this bound as yielding the reciprocal-sum conclusion.

2 · Going further

For a fixed integer $k \geq 3$ , let $r_{k} (N)$ be the largest size of a subset of ${1, \dots, N}$ containing no nonconstant $k$ -term arithmetic progression. Such a progression has the form $a, a + d, \dots, a + (k - 1) d$ with a nonzero common difference $d$ . The finite problem is to bound $r_{k} (N)$ as $N$ grows.

The manuscript gives the bound $r_{k} (N) \leq C_{k} N exp (- c_{k} (lo g N)^{ε_{k}})$ , where $C_{k}, c_{k}, ε_{k} > 0$ depend only on $k$ . Thus the estimate is stated separately for each fixed length, rather than with one set of constants for all lengths. The title describes these as quasipolynomial bounds for arithmetic progressions.

The associated infinite statement concerns a set $A$ of positive integers satisfying $\sum_{a \in A} 1/ a = \infty$ : it must contain arithmetic progressions of every finite length. The abstract presents the finite estimate as a quantitative strengthening of this conclusion, identified as Erdős’s conjecture. It does not describe the argument or the mechanism for obtaining the bound, so no proof strategy can be reported from the supplied text. Only the main manuscript is supplied; no companion-manuscript relationship is specified.

Useful notions

Positive integers and subsets
Equally spaced number sequences
Reciprocals and divergent sums
Logarithms and exponentials
Bounds with length-dependent constants

What was known before / what is new

Before

The source identifies the reciprocal-sum statement as Erdős’s conjecture. It supplies no earlier quantitative bounds or account of prior methods, so it does not support a more detailed historical comparison.

New

The manuscript states the reciprocal-sum conclusion for every finite progression length. It also supplies the displayed upper bound for $r_{k} (N)$ for every fixed $k \geq 3$ , with positive constants depending only on $k$ . The main result has a Lean formalization.

Limits and assumptions

The infinite conclusion assumes a divergent reciprocal sum; it makes no assertion here about sets whose reciprocal sum converges. The finite bound concerns avoidance of nonconstant progressions and is stated for each fixed $k \geq 3$ , with constants that may depend on $k$ ; their values are not supplied. The main result has a Lean formalization, but the supplied material does not identify its precise scope or establish that every accompanying claim is formalized. The abstract gives no proof details, companion manuscripts, or remaining open questions.

Mini glossary

Arithmetic progression: A sequence of numbers whose consecutive differences are all equal.
Nonconstant progression: An arithmetic progression whose common difference is nonzero.
Divergent reciprocal sum: A sum of the reciprocals of selected positive integers whose partial sums grow without bound.
$r_{k} (N)$: The largest size of a subset of ${1, \dots, N}$ containing no nonconstant arithmetic progression of length $k$ .
Lean formalization: A mathematical statement and its proof expressed and checked in the Lean proof assistant.

Keywords : Arithmetic progressions · Divergent reciprocal sums · Quasipolynomial bounds

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of ${1, \dots, N}$ with no nonconstant k-term progression has size at most $C_{k} N exp [- c_{k} (lo g N)^{ε_{k}}]$ , with positive constants depending only on k.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/159.md

Manuscripts in this family

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Quasipolynomial Bounds for Arithmetic Progressions

23/09/2026

To be verified

We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show $r_{k} (N) \leq C_{k} N exp (- c_{k} (lo g N)^{ε_{k}})$ with $C_{k}, c_{k}, ε_{k} > 0$ , where $r_{k} (N)$ is the largest size of a subset of ${1, \dots, N}$ with no nonconstant k-term arithmetic progression.

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