challenges / sum-difference-exponent-ii
Sum-difference exponent II (more differences than sums)
AlphaEvolve repository problem 43: find a finite set A of integers maximising log|A-A| / log|A+A|, the exponent C in |A-A| <= |A+A|^C. Scored against the best-known lower bound log(1+sqrt2)/log 2 = 1.27155 (Hennecart-Robert-Yudin simplex construction, a limit value no finite set is known to reach; AlphaEvolve got about 1.21).
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# Sum-difference exponent II: more differences than sums
## Goal
`construct.py` exposes `construct(max_size: int, time_budget: float, seed: int) -> list[int]`:
a list of **distinct integers** `A` with `2 <= len(A) <= max_size`. Maximise
C(A) = log|A-A| / log|A+A|
where `A+A = {a+b : a, b in A}` and `A-A = {a-b : a, b in A}`. This is problem 43 of the
AlphaEvolve repository of problems ("Sum-difference problem II", section 6.25 of *Mathematical
Exploration and Discovery at Scale*, arXiv:2511.02864): let `C` be the least constant with
`|A-A| <= |A+A|^C` for every finite `A ⊂ Z`; every explicit set is a lower bound for `C`.
Freiman-Pigarev / Ruzsa give `C <= 4/3`.
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