An estimate for positive definite functions on finite abelian groups and its applications
Lixia Wang, Ke Ye
TL;DR
This work establishes a sharp Fourier-analytic lower bound for the second largest value $\nu_2(f)$ of real-valued positive definite functions on a finite abelian group $G$, showing $\nu_2(f) \ge f(1) \left( 1 - \frac{ \pi^2 |G| }{2m^2 (|G| - 2^t (m-1)^{(s-t)/2})} \right)$. The authors develop a generalized covering argument connecting Rayleigh quotients to $\nu_2(f)$ and reinterpret the results through a graph-theoretic lens via weighted Cayley graphs. Three applications are then derived: (i) tight bounds on values of arbitrary functions on $G$, (ii) improved lower bounds for the relaxation and mixing times of random walks on $G$, with a quadratic enhancement in the relaxation time exponent, and (iii) new lower bounds on the size of sumsets $AB$ in finite abelian groups. The approach blends harmonic analysis with spectral graph theory to yield quantitative tools for additive combinatorics, random walks, and function estimation on finite groups, with explicit dependence on the Fourier support size and symmetry properties.
Abstract
This paper concentrates on positive definite functions on finite abelian groups, which are central to harmonic analysis and related fields. By leveraging the group structure and employing Fourier analysis, we establish a lower bound for the second largest value of positive definite functions. For illustrative purposes, we present three applications of our lower bound: (a) We obtain both lower and upper bounds for arbitrary functions on finite abelian groups; (b) We derive lower bounds for the relaxation and mixing times of random walks on finite abelian groups. Notably, our bound for the relaxation time achieves a quadratic improvement over the previously known one; (c) We determine a new lower bound for the size of the sumset of two subsets of finite abelian groups.