On the Sidon tails of $\left\{\lfloor x^n\rfloor\right\}$

TL;DR

This paper investigates the Sidon property of the tail sets of sequences formed by the integer parts of powers of real numbers, showing that for most x in (1,2), these sets are Sidon, but exceptions exist near 1 and 2.

Contribution

It proves that the tail sets of \\{loor{x^n}\\} are Sidon for almost all x in (1,2), and constructs specific examples near 1 and 2 where the sets are not Sidon.

Findings

Most tail sets are Sidon for x in (1,2)

Exceptions occur near x=1 and x=2

Existence of x and r where tail sets are not Sidon

Abstract

We prove that the tail of the sets $$\mathbf S_x := \big\{\left\lfloor x^n\right\rfloor : n\in \mathbb N\big\}$$ are Sidon for almost all $x\in (1,2)$. Then we prove that for all $\varepsilon>0$, there exists $x\in (1,\, 1+\varepsilon)$ and $r\in (2-\varepsilon,\, 2)$ such that $\mathbf S_x$ and $\mathbf S_r$ do not have a Sidon tail.