Structure and arithmetic in sets
Karsten Chipeniuk · cIRcle (University of British Columbia) · 2011
We prove results in arithmetic combinatorics involving sums of prime numbers and also some variants of the Erdös-Szemerédi sum-product phenomenon. In particular, we prove nontrivial lower bounds on the density in the integers of the sumset of a positive relative density subset of the primes. The proof of this result uses Green and Green-Tao pseudorandomness arguments to reduce the problem to an analogous statement for relatively dense subsets of the multiplicative subgroup of integers modulo a large integer N. The latter statement is resolved with a combinatorial argument which bounds high moments of a representation function. We also show that if two distinct sets A and B of complex numbers have very small productset, then they produce maximally large iterated sumsets. This uses an algebraic concept of the multiplicative dimension of a finite set. As an application of the case A=B, we obtain a quantitative version of a result of Chang on sums and products of distinct complex elements.