论文标题
套装的界限，几乎没有距离不同的模特一个理想的理想
Bounds for sets with few distances distinct modulo a prime ideal
论文作者
论文摘要
令$ \ mathcal {o} _k $为嵌入到$ \ mathbb {c} $中的代数数字字段$ k $的整数环。令$ x $为Euclidean Space $ \ Mathbb {r}^d $的子集，而$ d（x）$是$ x $中两个不同点的平方距离集。在本文中，我们证明，如果$ d（x）\ subset \ mathcal {o} _k $，并且存在$ s $ values $ a_1，\ ldots，a_s \ in \ in \ nathcal {o} _k $} _k $ nivters modulo a prime a Prime a Prime a prime $ $ \ mathfrak $ \ mathfrak of $ \ iis_; modulo $ \ mathfrak {p} $，$ d（x）$的每个元素与某些$ a_i $，然后$ | x |一致\ leq \ binom {d+s} {s}+\ binom {d+s-1} {s-1} $。
Let $\mathcal{O}_K$ be the ring of integers of an algebraic number field $K$ embedded into $\mathbb{C}$. Let $X$ be a subset of the Euclidean space $\mathbb{R}^d$, and $D(X)$ be the set of the squared distances of two distinct points in $X$. In this paper, we prove that if $D(X)\subset \mathcal{O}_K$ and there exist $s$ values $a_1,\ldots, a_s \in \mathcal{O}_K$ distinct modulo a prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ such that each $a_i$ is not zero modulo $\mathfrak{p}$ and each element of $D(X)$ is congruent to some $a_i$, then $|X| \leq \binom{d+s}{s}+\binom{d+s-1}{s-1}$.
