arXiv subjects

Zuxiang Kong

Publications and source records attributed to Zuxiang Kong.

2 recordsLinked to original sources

arXiv2026

Projection and fibering in groups of bounded exponent

We develop a projection and fibering method for sets of small combinatorial doubling in (not necessarily abelian) discrete groups. As an application, in the abelian case we prove that, if $A$ is finite, the ambient group has exponent $r$, and $|A+A|\leq K|A|$, then \[ |\langle A\rangle|\leq r^{(2+o(1))K}|A|. \] This answers a question of Ruzsa with an optimal leading coefficient, independent of Fox--Pham. The main ingredient is a discrete version of a fiber spillover argument. For sets in $2$-step nilpotent groups of exponent $r$, we also prove that $|A^3|\leq K|A|$ implies $|\langle A\rangle|\leq r^{(2+o_K(1))K}|A|$. The proof combines the abelian theorem with a weighted averaging of central fibers and commutators.

math.NT↗

arXiv2024

Measure doubling in unimodular locally compact groups and quotients

We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $μ_G$, and a compact $A\subseteq G$ of positive measure with $μ_G(A^2)\leq Kμ_G(A)$. Let $H$ be a closed normal subgroup of G and $π: G \rightarrow G/H$ be the quotient map. With the further assumption that $A= A^{-1}$, we show $$μ_{G/H}(πA ^2) \leq K^2 μ_{G/H}(πA).$$ We also demonstrate that $K^2$ cannot be replaced by $(1-ε)K^2$ for any $ε>0$. In the general case (without $A=A^{-1}$), we show $μ_{G/H}(πA ^2) \leq K^3 μ_{G/H}(πA)$, improving an earlier result by An, Jing, Zhang, and the third author. Moreover, we are able to extract a compact set $B\subseteq A$ with $μ_G(B)> μ_G(A)/2$ such that $ μ_{G/H}(πB^2) < 2K μ_{G/H}(πB)$.

math.GR↗