Common difference sets, higher order recurrence sets and related problemsAlgebra & Discrete Mathematics
|Wen Huang, University of Science and Technology of China
|Tue, Aug 2 2016, 1:10PM
It is believed that a “big” subset of natural numbers should contain “good” linear structures, for example arbitrarily long arithmetic progressions (AP). Szemerdi's Theorem asserts that every positive upper Banach density subset of natural numbers has this property. In this talk, we will consider the structure of all common differences of AP with length k+1 appeared in a “big” subset of natural numbers. By the Furstenberg correspondence principle, the common difference set of AP of a syndetic (or positive upper Banach density) subset are related to the higher order recurrence sets in dynamical systems. It is shown that a higher order recurrence sets is an almost Nil Bohr-sets, and Nil Bohr-sets could be characterized via generalized polynomials. Hence the common difference set of AP of a syndetic (or positive upper Banach density) subset is an almost Nil Bohr-sets, which could be characterized via generalized polynomials. Some related problems in dynamical system (for example multiple ergodic average problem) will be also discussed. Finally we will review some progress on the difference set of primes.