Hypercontractivity of spherical averages in Hamming space

TL;DR

This paper establishes a dimension-independent bound on the $L_p o L_2$ norm of spherical averaging operators in Hamming space, revealing new hypercontractivity properties and implications for set structure in boolean analysis.

Contribution

It provides the first dimension-independent $L_p o L_2$ bound for Hamming sphere averaging operators, extending classical hypercontractivity results.

Findings

Bound on $L_p o L_2$ norm with $p=1+(1-2 ext{radius})^2$

Eigenvalue analysis of Hamming sphere operators

Application to structure of sets with large sumsets

Abstract

Consider the linear space of functions on the binary hypercube and the linear operator $S_\delta$ acting by averaging a function over a Hamming sphere of radius $\delta n$ around every point. It is shown that this operator has a dimension-independent bound on the norm $L_p \to L_2$ with $p = 1+(1-2\delta)^2$. This result evidently parallels a classical estimate of Bonami and Gross for $L_p \to L_q$ norms for the operator of convolution with a Bernoulli noise. The estimate for $S_\delta$ is harder to obtain since the latter is neither a part of a semigroup, nor a tensor power. The result is shown by a detailed study of the eigenvalues of $S_\delta$ and $L_p\to L_2$ norms of the Fourier multiplier operators $\Pi_a$ with symbol equal to a characteristic function of the Hamming sphere of radius $a$ (in the notation common in boolean analysis $\Pi_a f=f^{=a}$, where $f^{=a}$ is a degree-$a$ component of function $f$). A sample application of the result is given: Any set $A\subset \FF_2^n$ with the property that $A+A$ contains a large portion of some Hamming sphere (counted with multiplicity) must have cardinality a constant multiple of $2^n$.