On Bipartite Distinct Distances in the Plane

Abstract

Given sets P, Q ⊆ R 2 of sizes m and n respectively, we are interested in the number of distinct distances spanned by P × Q. Let D(m, n) denote the minimum number of distances determined by sets in R 2 of sizes m and n respectively, where m ≤ n.In this work, we show that Elekes' construction is tight by deriving the lower bound of D(m, n) = Ω( √ mn) when m ≤ n 1/3 . This is done by adapting Székely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of D(m, n) = Ω( √ mn/ log n) when m ≥ n 1/3 .

Distinct Distances Between a Circle and a Generic Set

“…Finally, Mathialagan [12] extended these results in R 2 to the setting where P and P 1 are both unrestricted point sets (of size m and n respectively)…”

“…The relationship between Mathialagan [12] and our work is a little more complicated. Mathialagan's results are much more general than ours, as neither of the sets is restricted at all.…”

“…We prove the number of distinct distances between points in S and points in P is the minimum of |S| 8 9 |P | 2 9 , |P | 2 and |S| 2 . This builds on work of Pach and De Zeeuw [15], Bruner and Sharir [3], McLaughlin and Omar [13] and Mathialagan [12] on distances between pairs of sets.…”

Distance problems for planar hypercomplex numbers

“…We denote the number of such distinct distances by D(P 1 , P 2 ). The following bound was derived in [18]. Theorem 2.5.…”

“…Recently, Guth and Katz [9] introduced novel polynomial methods to derive the bound Ω n log n , which matches Erdős's conjecture up to a factor of √ log n. The distinct distances problem also has a large number of variants, and some of the main variants remain wide open. For example, the problem has been studied in higher dimensions [23], in finite fields [3,12,16,20], and with bipartite distances [18]. For more information, see this book about the problem [7] and a survey of open distinct distances problems [21].…”