Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets

TL;DR

This paper explores the approximation properties of points in minimal nilsystems, linking topological dynamics with multiplicative combinatorics, and extends classical recurrence and van der Waerden-type theorems.

Contribution

It establishes that in minimal nilsystems, points approximate dense sets under restricted powers, connecting this to multiplicative thickness of return-time sets and generalizing multiple recurrence results.

Findings

Points in minimal nilsystems approximate dense sets under restricted powers

Nil-Bohr sets and typical return-time sets are multiplicatively thick in cosets

The results generalize Furstenberg-Weiss recurrence and van der Waerden's theorem

Abstract

In the topological dynamical system $(X,T)$, a point $x$ simultaneously approximates a point $y$ if there exists a sequence $n_1$, $n_2$, ... of natural numbers for which $T^{n_i} x$, $T^{2n_i}x$, ..., $T^{k n_i} x$ all tend to $y$. In 1978, Furstenberg and Weiss showed that every system possesses a point which simultaneously approximates itself (a multiply recurrent point) and deduced refinements of van der Waerden's theorem on arithmetic progressions. In this paper, we study the denseness of the set of points that are simultaneously approximated by a given point. We show that in a minimal nilsystem, all points simultaneously approximate a $\delta$-dense set of points under a necessarily restricted set of powers of $T$. We tie this theorem to the multiplicative combinatorial properties of return-time sets, showing that all nil-Bohr sets and typical return-time sets in a minimal system are multiplicatively thick in a coset of a multiplicative subsemigroup of the natural numbers. This yields an inhomogeneous multiple recurrence result that generalizes Furstenberg and Weiss' theorem and leads to new enhancements of van der Waerden's theorem. This work relies crucially on continuity in the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, and Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye.