Denote by $f_D(n)$ the maximum size of a set family $\mathcal{F}$ on $[n] \stackrel{\mbox{\normalfont\tiny def}}{=} \{1, \dots, n\}$ with distance set $D$. That is, $|A \bigtriangleup B| \in D$ holds for every pair of distinct sets $A, B \in \mathcal{F}$. Kleitman's celebrated discrete isodiametric inequality states that $f_D(n)$ is maximized at Hamming balls of radius $d/2$ when $D = \{1, \dots, d\}$. We study the generalization where $D$ is a set of arithmetic progression and determine $f_D(n)$ asymptotically for all homogeneous $D$. In the special case when $D$ is an interval, our result confirms a conjecture of Huang, Klurman, and Pohoata. Moreover, we demonstrate a dichotomy in the growth of $f_D(n)$, showing linear growth in $n$ when $D$ is a non-homogeneous arithmetic progression. Different from previous combinatorial and spectral approaches, we deduce our results by converting the restricted distance problems to restricted intersection problems. Our proof ideas can be adapted to prove upper bounds on $t$-distance sets in Hamming cubes (also known as binary $t$-codes), which has been extensively studied by algebraic combinatorialists community, improving previous bounds from polynomial methods and optimization approaches.

翻译：记 $f_D(n)$ 为定义在 $[n] \stackrel{\mbox{\normalfont\tiny def}}{=} \{1, \dots, n\}$ 上且具有距离集 $D$ 的集合族 $\mathcal{F}$ 的最大规模。即对于任意两个不同的集合 $A, B \in \mathcal{F}$，均有 $|A \bigtriangleup B| \in D$。Kleitman 著名的离散等周不等式指出，当 $D = \{1, \dots, d\}$ 时，$f_D(n)$ 在半径为 $d/2$ 的汉明球上取得最大值。本文研究 $D$ 为等差数列集合的推广情形，并对所有齐次 $D$ 渐近地确定了 $f_D(n)$。特别地，当 $D$ 是一个区间时，我们的结果证实了 Huang、Klurman 和 Pohoata 的猜想。此外，我们揭示了 $f_D(n)$ 增长中的二分现象：当 $D$ 为非齐次等差数列时，其关于 $n$ 呈线性增长。与以往的组合和谱方法不同，我们通过将受限距离问题转化为受限交集问题来推导结论。我们的证明思路可推广至汉明立方（亦称二元 $t$ 码）中 $t$-距离集的上界估计，该问题在代数组合学界被广泛研究，本文结果改进了先前基于多项式方法和优化途径的界限。