Counting Exceptional Points for Rational Numbers Associated to the Fibonacci Sequence

TL;DR

This paper explores the properties of the $t$-metric Mahler measure for algebraic numbers related to Fibonacci numbers, proposing a conjecture that suggests the existence of rational numbers with infinitely many exceptional points.

Contribution

The paper formulates a Fibonacci-based conjecture on the $t$-metric Mahler measure and proves its optimality, linking it to the existence of rational numbers with arbitrarily many exceptional points.

Findings

Proposes a Fibonacci-inspired conjecture on $t$-metric Mahler measures.

Shows the conjecture's validity would imply infinitely many exceptional points for some rationals.

Resolves special cases of the conjecture through computational methods.

Abstract

If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(\alpha)$. For fixed $\alpha\in \bar{\mathbb Q}$, the map $t\mapsto m_t(\alpha)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $\alpha$. It remains open to determine whether there is a sequence of elements $\alpha_n\in \bar{\mathbb Q}$ such that the number of exceptional points for $\alpha_n$ tends to $\infty$ as $n\to \infty$. We utilize a connection with the Fibonacci sequence to formulate a conjecture on the $t$-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.