Few distinct distances implies no heavy lines

11/17/2015 - 12:15pm
11/17/2015 - 1:10pm
Adam Sheffer (CalTech)

After Guth and Katz's almost tight bound for the distinct distances problem, one of the main open problems is to characterize the structure of planar point sets that determine a small number of distinct distances. We show that if a set of n points determines o(n) distinct distances then no line contains n^{7/8} points of the set and no circle contains n^{5/6} such points. Our analysis combines tools from incidence geometry and additive combinatorics. In the talk, before getting to our result I will spend some time on surveying the distinct distances problem in general. Joint work with Joshua Zahl and Frank de Zeeuw.

Millikan 2099, Pomona College