Universal Fourier-inertia bounds for prescribed even distances

Abstract

The study of set families with restricted Hamming distances is a classical topic of extremal combinatorics and coding theory. Let \(H=\{A\subseteq[n]: |A|\text{ is even}\}\) be the even subcube. Let \(\ell_1,\dots,\ell_t\) be distinct positive integers and set \(\mathcal L=\{2\ell_1,\dots,2\ell_t\}\). We prove that, for all sufficiently large \(n\), every family \(\mathcal F\subseteq H\) satisfying \( |A\triangle B|\in\mathcal L \) for all \(A\ne B\in\mathcal F\) has \[ |\mathcal F|\le \sum_{i=0}^{t}\binom{n-1}{i}. \] This is best possible as a universal bound, with equality attained at the distance set \(\mathcal L=\{2,4,\dots,2t\}\). Our proof uses a Fourier-inertia argument based on a universal low/high boundary-layer sign pattern for the Fourier transform of the distance-polynomial kernel on the even subcube: the prescribed distances enter only through lower-order Fourier terms, while the leading boundary-layer signs depend solely on \(t\). This even-subcube result immediately yields an odd-subcube analogue and, through parity reductions, provides bounds for arbitrary distance sets. In particular, this approach recovers the classical interval bounds of Kleitman and the corresponding interval bounds of Huang--Klurman--Pohoata, while offering a direct spectral proof of the maximality of \(\{2,4,\dots,2t\}\) among all fixed \(t\)-distance sets.