Quasipolynomial Bounds for Arithmetic Progressions
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.
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
any set of positive integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions, with quasipolynomial quantitative bounds in Szemerédi’s theorem. Sidorenko’s conjecture. I described Sidorenko’s conjecture , which predicts that a fixed bipartite graph has at least the…