An explicit upper bound for the Helfgott delta in SL(2,p)

TL;DR

This paper establishes an explicit upper bound for Helfgott's delta in SL(2,p), narrowing the range of possible values and providing evidence that the true delta might be close to this upper limit.

Contribution

The paper provides a new explicit upper bound for Helfgott's delta in SL(2,p), improving understanding of growth in finite groups.

Findings

Upper bound for delta is approximately 0.3012

Evidence suggests the true delta may be close to the upper bound

Improves previous lower bound of 1/3024

Abstract

Helfgott proved that there exists a $\delta>0$ such that if $S$ is a symmetric generating subset of $SL(2,p)$ containing 1 then either $S^3=SL(2,p)$ or $|S^3|\geq |S|^{1+\delta}$. It is known that $\delta\geq 1/3024$. Here we show that $\delta\leq(\log_2(7)-1)/6 \approx 0.3012$ and we present evidence suggesting that this might be the true value of $\delta$.