Abstract:We show that very small distances in a planar integral point set are essentially one-dimensional. Let P be a non-collinear set of n points in the plane, all of whose pairwise distances are integers. We prove that, for a sufficiently small $c>0$, at most one pair of points can determine a distance below $n^{c\log\log{n}}$ unless all such short pairs are supported on a single line. We also give a construction showing that the line-supported alternative is necessary: there are arbitrarily large non-collinear integral point sets with one off-line point and with the first two distinct distances 2 and 4. We conjecture that in the general case (no three points are collinear) even the smallest distance should be large, at least $n^{c\log\log{n}}$, however we can prove a linear lower bound only.
From: Jozsef Solymosi [view email]
[v1]
Wed, 24 Jun 2026 18:53:11 UTC (9 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。