{"id":"palomar/PALOMAR-2026-09-21-000004","url":"https://github.com/JD-Jones-ASES/nk-lean/tree/55acbf0def926afb15eb243a3c3f2a3b42439bce","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-21-000004-v1.json","repo":"https://github.com/JD-Jones-ASES/nk-lean","tags":["math.NT","math.CO","msc:11B30","msc:11B75","msc:05C20","number-theory","additive-combinatorics","perfect-powers"],"score":0,"title":"Lower bounds for sets without perfect-power differences","trust":"high","domain":"math.NT","source":"palomar","authors":["JD Jones"],"reviews":0,"subject":"math.NT","summary":"Let D_k(N) be the largest size of a subset of {1, ..., N} with no positive k-th-power difference. We prove D_k(N) >= c_k N^alpha_k for every N >= 1, with positive constants c_k and exponents 0.75806770413, 0.9142, and 0.95295 for squares, fourth powers, and sixth powers. The proofs use a general interval-moment criterion for every positive integer k. We also prove arithmetic lifting results, exact product and prime-depth laws for interval capacity, and existence of the binary square capacity limit. The square construction refines Naslund's published example while retaining its binary…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-21","reactions":{},"formalized":true,"formalizes":[{"title":"Square-difference-free sets of exponent 0.75806746","identifier":"","relationship":"adapts"},{"title":"Square-Difference-Free Sets beyond the Three-Quarter Barrier","identifier":"arXiv:2608.01325","relationship":"adapts"}]}