Universal Fourier-inertia bounds for prescribed even distances

The study of set families with restricted Hamming distances is a classical topic of extremal combinatorics and coding theory. Let $H=\{A\subseteq [n]:\mathrm{\mid }A\mathrm{\mid }\text{ is even}\}$ be the even subcube. Let ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_t$ be distinct positive integers and set $\mathcal{L}=\{2{\mathrm{\ell }}_1,\dots ,2{\mathrm{\ell }}_t\}$. We prove that, for all sufficiently large $n$, every family $\mathcal{F}\subseteq H$ satisfying $\mathrm{\mid }A\mathrm{△}B\mathrm{\mid }\in \mathcal{L}$ for all $A\mathrm{\ne }B\in \mathcal{F}$ has $$\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le \sum _{i=0}^t\left(\genfrac{}{}{0px}{}{n-1}{i}\right)\mathrm{.}$$ 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.