Uniform dilations modulo a prime
26/02/2023
How large must k=k(epsilon,p) be, so that every set X of k integers in [1,p] must have a dilation nX modulo p, where all gaps are at most of length epsilon*p?
How large must k=k(epsilon,p) be, so that every set X of k integers in [1,p] must have a dilation nX modulo p, where all gaps are at most of length epsilon*p?
Given a lacunary sequence of integers, Erdos asked for the chromatic number of the graph on the integers where two integers are linked by an edge is their difference is in the sequence. determining the optimal bound for this chromatic number is closely related to a question on Diophantine approximation considered by Khinchin, Erdos and Katznelson.
Glasner showed that every infinite set on the circle, has a dilation that is epsilon-dense. What is the sharp quantitative version of his result?