Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds. 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 $\{1,\ldots,N\}$ with no nonconstant k-term progression has size at most $C_kN\exp[-c_k(\log N)^{\varepsilon_k}]$, with positive constants depending only on k.
released 2026-09-23 | 7 theorems · 69 lemmas · 104 proofs · 104,502 words |
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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
$\displaystyle r_k(N)\le C_kN\exp\bigl(-c_k(\log N)^{\varepsilon_k}\bigr)$
with $C_k,c_k,\varepsilon_k\gt 0$, where $r_k(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ with no nonconstant k-term arithmetic progression.