challenges / furstenberg-sarkozy
Furstenberg–Sárközy: modular sets avoiding k-th power differences
AlphaEvolve Problem 31: pick a squarefree modulus m and a subset A of Z/mZ in which no two elements differ by a nonzero k-th power residue; the density exponent log|A|/log m lifts to a lower bound for the largest square-difference-free (k=2) or cube-difference-free (k=3) subset of {1..N}. Scored against Lewko's 12 elements in Z/205Z and 14 elements in Z/91Z.
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# Furstenberg–Sárközy: modular sets avoiding k-th power differences
## Goal
The Furstenberg–Sárközy theorem says a subset of `{1, ..., N}` with no two elements differing by a
perfect square has size `o(N)`. The best lower bounds come from a modular construction (Ruzsa
1984): take a squarefree modulus `m` and a set `A ⊆ Z/mZ` in which no two distinct elements differ
by a nonzero `k`-th power residue mod `m`; writing integers in base `m` with digits from `A`
lifts this to a subset of `{1, ..., N}` of size about `N^q` with `q = log|A| / log m` (Ruzsa's
exponent for squares is then `½(1 + q)`). Ruzsa used `m = 65`, `|A| = 7`; Lewko (2015) found
`m = 205`, `|A| = 12`, which is the current record for squares. For cubes the known construction
is 14 elements in `Z/91Z`.
This is AlphaEvolve Problem 31 (Section 6.16 of arXiv:2511.02864; the notebook header numbers it
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