Few distinct distances implies no heavy lines

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

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.

Where: 
Millikan 2099, Pomona College