An ergodic approach to equations of the form $x+y=\lfloor \alpha (n)\rfloor$

Abstract

Motivated by questions and results of Erdős, Sárközy and Sós (1989), and Khalfalah and Szemerédi (2006), we introduce an ergodic approach to additive problems concerning solutions of equations of the form \[ x+y=\lfloorα(n)\rfloor,\quad x,y\in A,n\in\mathbb{N} \] where $\alpha (t)$ is a sufficiently regular function, such as a polynomial or a logarithmico-exponential function of polynomial growth (e.g. $\alpha (t)=t^{5/2}+t^2\log t$), and $A$ is either a given set of positive upper density, or a cell of a prescribed finite partition of $\mathbb{N}$. We give a complete description of the Hardy field functions of polynomial growth for which these partition and density problems are always solvable. The ergodic approach also applies to the more general problem of finding $x\in A_1,y\in A_2$, where $A_1,A_2$ are two given sets of positive density. This in turn allows us to treat equations of the form $kx+ly=\lfloor \alpha (n)\rfloor$, where $k,l\in \mathbb{Z}$. Some of our results also hold true when $A$ is a subset of the primes with positive relative upper density.

69 pages