{"id":"palomar/PALOMAR-2026-09-19-000006","url":"https://github.com/enaslund/sarkozy-lower-bound-0.758/tree/5a83f4174578c242ff3ea9373c2d4b9e406a9ffb","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-19-000006-v2.json","repo":"https://github.com/enaslund/sarkozy-lower-bound-0.758","tags":["math.NT","math.CO","msc:11B75","msc:05D05","additive-combinatorics","number-theory","sarkozy-theorem"],"score":0,"title":"An interval-moment criterion for square-difference-free integer sets","trust":"high","domain":"math.NT","source":"palomar","authors":["Eric Naslund"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-09","reactions":{},"formalized":true,"formalizes":[{"title":"Ranked residues, intervals, and the integer construction","identifier":"","relationship":"adapts"},{"title":"Square-Difference-Free Sets beyond the Three-Quarter Barrier","identifier":"https://arxiv.org/abs/2608.01325v1","relationship":"adapts"}]}