Density of oscillating sequences in the real line

Abstract: In this talk, we study the density properties in the real line of oscillating sequences of the form $( g(k) \cdot F(ka) )_{k \in \mathbb{N}}$, where $g$ is a positive increasing function and $F$ a real continuous $1$-periodic function. This extends work by Berend, Boshernitzan and Kolesnik who established differential properties on the function $F$ ensuring that the oscillating sequence is dense modulo $1$. More precisely, when F has finitely many roots in $[0,1)$, we provide necessary and sufficient conditions for the oscillating sequence under consideration to be dense in $\mathbb{R}$. All the related results are stated in terms of the Diophantine properties of $a$, with the help of the theory of continued fractions.

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar