Submit progress

on Unnormalized single-set sum-difference exponent ($C_{3e}$)

The statement

For a finite non-empty set $A$ of integers, $C_{3e}$ is the least constant such that

|A - A| \le |A + A|^{C_{3e}}

for every such $A$ ; equivalently, $C_{3e} = \sup_A \log\lvert A-A\rvert / \log\lvert A+A\rvert$ . This is Problem 6.43 of [GGSWT2025].

It measures how much larger the difference set can be than the sumset, without normalizing by $\lvert A\rvert$ — which is what distinguishes it from $C_{3d}$ , the least $\theta$ with $\lvert A+A\rvert/\lvert A\rvert \le (\lvert A-A\rvert/\lvert A\rvert)^{\theta}$ . The two normalizations have different classical bounds and different records, and conflating them has caused published confusion; see the caution below.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.