challenges / sum-difference-exponent-i
Sum-difference exponent I (more sums than differences)
AlphaEvolve repository problem 42: find a finite set A of integers maximising log(|A+A|/|A|) / log(|A-A|/|A|), the exponent C in |A+A|/|A| <= (|A-A|/|A|)^C. Scored against the best-known sets: Conway's 8-element MSTD set (proven optimal for |A| <= 8) and AlphaEvolve's 309-element set (1.12194, open).
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# Sum-difference exponent I: more sums than differences
## 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| / |A|) / log(|A-A| / |A|)
where `A+A = {a+b : a, b in A}` and `A-A = {a-b : a, b in A}`. This is problem 42 of the
AlphaEvolve repository of problems ("Sum-difference problem I", section 6.25 of *Mathematical
Exploration and Discovery at Scale*, arXiv:2511.02864): let `C` be the least constant with
`|A+A|/|A| <= (|A-A|/|A|)^C` for every finite `A ⊂ Z`; every explicit set is a lower bound for `C`.
Sets with `|A+A| > |A-A|` (MSTD sets, "more sums than differences") are exactly the sets scoring
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