A long gap in a dilate modulo a prime

Slug: long-gap-dilate
List: P08
Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
Status: open
Area: Additive combinatorics
Sources: Green 100 #32; Shakan, SIAM J. Discrete Math. 34 (2020), arXiv:2004.14828 (the list id arXiv:2205.14038 is a different paper)
Started: 2026-08-17

Statement

Let $p$ be prime and $A\subset\mathbb Z/p\mathbb Z$ . Write $g(A)$ for the longest run of consecutive residues missed by $A$ , and $d\cdot A=\{da:a\in A\}$ . Green #32: if $|A|=\sqrt p$ , is there a dilate with $g(d\cdot A)\ge 100\sqrt p$ ?

Shakan (2020): for every $A$ with $|A|>1$ ,

\sup_{d\in\mathbb F_p^\times} g(d\cdot A)\;\ge\; 2p/|A|-2.

At $|A|\sim\sqrt p$ this is a gap of $2\sqrt p-2$ . Green records this as the limit of the polynomial method.

Tonight

A certified improvement of the universal constant 2, or a documented incomplete search with replayable compute. Small-prime extremizer tables are an incomplete search unless they imply a larger universal $C$ .

Outcome

Incomplete search. Shakan’s 2 is still the published leading constant. Exact $G(p,\mathrm{round}\sqrt p)$ now includes $G(73,9)=24$ (Cadical unsat at $T=24$ , Glucose4 witness at $T=25$ ). Upper bounds $G(79,9)\le 26$ , $G(83,9)\le 27$ , $G(89,9)\le 30$ have no certified floors. The search in compute/q1/ saturates the rising-factorial Alon degrees and finds no $C\to 2$ family through $p=199$ . One new exact row is not a universal $C>2$ . Residue, not a dent.