Two-step product growth in SL_2(F_p)

Definitions and mathematical statement

Brought by B. Green and originally stated by A. Venkatesh, the source asks whether $A\subseteq\operatorname{SL}_2({\mathbb{F}}_p)$ with $|A|\sim p^{5/2}$ must satisfy $|AA|>p^{5/2+\delta}$ for some positive fixed $\delta$. Here $AA=\{ab:a,b\in A\}$. To make the size condition explicit, the aid's result applies whenever $c_1p^{5/2}\leq|A|\leq c_2p^{5/2}$ with fixed positive comparability constants, and therefore also under the stronger ratio-one meaning of $\sim$.

The summary attributes the affirmative answer to the Babai--Nikolov--Pyber two-step product-mixing theorem. Its reported quantitative form is: if a finite group $G$ has order $N$ and every nontrivial irreducible complex representation has dimension at least $D$, then every pair of nonempty sets $A,B\subseteq G$ satisfies \[ |AB|\geq\frac{N}{1+\dfrac{N^2}{D|A||B|} (1-|A|/N)(1-|B|/N)}. \] No symmetry, identity membership or independent generating hypothesis on these subsets is required by this stated bound.

For $G=\operatorname{SL}_2({\mathbb{F}}_p)$, $|G|=p(p^2-1)$ and the supplied representation-degree lower bound is $(p-1)/2$ for nontrivial complex irreducibles. Applying the product bound with $B=A$ and the lower size comparison gives \[ |AA|\gg_{c_1}p^3. \] Consequently, for every fixed $0<\delta<1/2$, in particular $\delta=1/4$, the requested strict inequality holds for all sufficiently large primes, with the threshold allowed to depend on $c_1$ and $\delta$. This is a full answer to the source's fixed-positive-saving question. The endpoint $\delta=1/2$ with its literal strict comparison to $p^3$ is impossible since the entire group has only $p^3-p$ elements.

The summary says its fuller artifacts contain a noncommutative Fourier proof of the product estimate and an elementary unipotent-character proof of the representation bound. Those proofs are not supplied in this registry snapshot and are not independently certified by this statement-only draft.

Short English statement

The supplied product-mixing theorem gives a positive proportion of the whole group after only two products. That is much larger than the input scale and proves every fixed exponent saving below one half for sufficiently large primes. The named theorem and representation bound remain attributed to the supplied report.

Sources

• Local UnsolvedMath registry, permanent ID $20001116$, source code catalogue entry, supplied complete source record and research summary.
• American Institute of Mathematics, Recent trends in additive combinatorics, Problem 3.4, brought by B. Green and originally stated by A. Venkatesh; supplied URL https://aimath.org/WWN/additivecomb/additivecomb.pdf. Canonical local record aim-combinatorics-notes.json, notes[240]. The supplied verification reports a direct PDF inspection; none was performed here.
• Babai--Nikolov--Pyber two-step product-mixing theorem, as named by the research summary. No full title, year or bibliographic entry is supplied, nor a full authoritative citation for the representation-degree estimate or stated refinement. The original literature field is empty.
• Research attempt 1: full solution; problem solved at the recorded run and classified SOLVED-IN-LITERATURE; supplied rigor label "proved", novelty confidence "low". The machine-generated aid is not peer reviewed and requires independent verification before citation.