Arithmetic properties of generalised polynomials

We study generalised polynomials, that is, functions that can be expressed using standard algebraic operations and the floor function. Generalised polynomials have long been studied (both explicitly and implicitly) in number theory and dynamics, often using dynamical and ergodic-theoretic methods (especially dynamics on nilmanifolds). These methods enable one to deduce precise information about the average behaviour of generalised polynomials, but allow for complicated behaviour on special sets of density zero. We will present some of the classical results and give examples of various interesting arithmetic and combinatorial behaviour occurring along sets of density zero as well as restrictions to what is possible. We will also state an analogue of Hadamard's quotient theorem for generalised polynomials (the classical version concerns linear recurrences). We will also formulate several open problems. The talk is based on joint work with Jakub Konieczny (Oxford).