Counterexamples to integer-coefficient criteria for recurrence along functions from a Hardy field

TL;DR

This paper provides counterexamples showing that certain integer-coefficient criteria do not guarantee recurrence properties along functions from a Hardy field, challenging previous assumptions in the field.

Contribution

It constructs explicit counterexamples demonstrating the failure of recurrence criteria based on integer derivatives for functions in a Hardy field.

Findings

Every function in the constructed class has limit zero or infinity.

There exists a set of positive density with non-thick common return times.

Common return-time sets can be empty despite full integer derivative-span conditions.

Abstract

We give negative answers to two questions of Bergelson, Moreira, and Richter concerning recurrence along functions from a Hardy field. For the pair \(f_1(t)=t^{3/2}\) and \(f_2(t)=\lambda t^{3/2}+t\), where \(\lambda\in\mathbb R\setminus\mathbb Q\), singled out in their integer-coefficient derivative-span question, we prove that every \(F\in\nablaz(f_1,f_2)\) satisfies \(\lim_{t\to\infty}|F(t)|\in\{0,\infty\}\). Nevertheless, there is a set \(E\subset\mathbb N\) of positive natural density such that \(R_{f_1}(E)\cap R_{f_2}(E)\) is piecewise syndetic and not thick. Thus the proposed integer-coefficient replacement does not imply thickness. We further show that, even under the same full integer derivative-span condition, the common return-time set may be empty. This stronger obstruction also gives a negative answer to their question asking whether the recurrence conclusion of Theorem A follows from joint intersectivity of the integer polynomials in \(\operatorname{poly}(f_1,\ldots,f_k)\). The constructions use elementary Bohr sets.