Non-convex geometry of numbers and continued fractions

TL;DR

This paper extends the concept of $p$-continued fractions to the non-convex region $0<p<1$, establishing bounds on approximation coefficients and analyzing the behavior of convergents in this regime.

Contribution

It generalizes $p$-continued fractions to non-convex $L^p$ quasinorms and provides bounds on approximation coefficients as well as convergence properties.

Findings

Approximation coefficients are bounded by $1/\sqrt{5}+\varepsilon_p$ with $\varepsilon_p \to 0$ as $p \to 0$.

The bounds are sharp in the limit, aligning with Hurwitz's theorem.

Analyzes the maximum number of skipped regular convergents in the $p$-continued fraction.

Abstract

In recent work, the first two authors constructed a generalized continued fraction called the $p$-continued fraction, characterized by the property that its convergents (a subsequence of the regular convergents) are best approximations with respect to the $L^p$ norm, where $p\geq 1$. We extend this construction to the region $0<p<1$, where now the $L^p$ quasinorm is non-convex. We prove that the approximation coefficients of the $p$-continued fraction are bounded above by $1/\sqrt{5}+\varepsilon_p$, where $\varepsilon_p\to 0$ as $p\to 0$. In light of Hurwitz's theorem, this upper bound is sharp, in the limit. We also measure the maximum number of consecutive regular convergents that are skipped by the $p$-continued fraction.