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An interval-moment criterion for square-difference-free integer sets

Eric Naslund

Palomar Registrypalomar/PALOMAR-2026-09-19-000006Oct 9, 2026math.NTscore 0

We prove a general interval-moment criterion for square-difference-free integer sets and apply it to obtain an improved unconditional lower bound in Sarkozy's theorem, with exponent 0.7580758318008816. If D(N) is the largest size of a subset of {1, ..., N} containing no two distinct elements whose difference is a square, then for every epsilon > 0 and all sufficiently large N we prove D(N) >= N^(0.7580758318008816 - epsilon). The exact exponent is 473797394875551/625000000000000. The reusable criterion starts from finite digit sets in pairwise coprime square bases, with ordered intervals and…

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