Non-convex geometry of numbers and continued fractions

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 $p\ge 1$. 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 $1/\sqrt{5}+{\epsilon }_p$, where ${\epsilon }_p\to 0$ 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.

18 pages, 3 figures