Few distinct distances implies no heavy lines or circles

Abstract

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains Ω(n 7/8 ) points of P and no circle contains Ω(n 5/6 ) points of P.We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in [19] for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang [20].A significant difference between our approach and that of [19] (and of other related results) is that instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.

Bisector Energy and Few Distinct Distances

“…This problem was recently approached from the other direction in [13,14,18]. Combining the results of these three papers implies the following.…”

“…Recently, it was noticed that this bound can be slightly improved to Ω(log n) points on a line (see [17]). Assuming that no line contains an asymptotically larger number of points, one can prove the existence of Ω(n/ log n) distinct lines that contain Ω(log n) points of P. By inspecting Szemerédi's proof, it is also apparent that these lines are perpendicular bisectors of pairs of points of P.…”

Distinct distances on algebraic curves in the plane

“…In [19], Sharir and Solymosi show, using a method based on that of [18], that between three non-collinear points and n other points there are Ω(n 6/11 ) distinct distances. In [20], Sheffer, Zahl, and De Zeeuw extend the method of [18] to the case where one set of points in R 2 is constrained to a line, while the other is unconstrained. Finally, in [16], Raz, Sharir, and Solymosi use the approach of [18] to improve the bounds in the more general problem of [7].…”

A Survey of Elekes-Rónyai-Type Problems

“…Another variant of Theorem 3.1 was studied by Sheffer, Zahl, and De Zeeuw [49]: Suppose that P 1 is contained in a curve C, and P 2 is an arbitrary set in R 2 . In general it is difficult to obtain bounds in this situation, but [49] showed that if C is a line or a circle, then the number of distances determined by P 1 ∪ P 2 is reasonably large.…”

“…Another variant of Theorem 3.1 was studied by Sheffer, Zahl, and De Zeeuw [49]: Suppose that P 1 is contained in a curve C, and P 2 is an arbitrary set in R 2 . In general it is difficult to obtain bounds in this situation, but [49] showed that if C is a line or a circle, then the number of distances determined by P 1 ∪ P 2 is reasonably large. This result was used to show that a point set that determines o(n) distinct distances cannot have too many points on a line or a circle, which is a small step towards the conjecture of Erdős that a set of n points with o(n) distinct distances must resemble an integer grid.…”