Bounds for sets with few distances distinct modulo a prime ideal
Hiroshi Nozaki ยท Algebraic Combinatorics ยท 2023
Let ๐ช K be the ring of integers of an algebraic number field K embedded into โ . Let X be a subset of the Euclidean space โ 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 ) โ ๐ช K and there exist s values a 1 , ... , a s โ ๐ช K distinct modulo a prime ideal ๐ญ of ๐ช K such that each a i is not zero modulo ๐ญ and each element of D ( X ) is congruent to some a i , then | X | โค d + s s + d + s - 1 s - 1 .