Non-convex geometry of numbers and continued fractions (Q6187376)

This paper is a sequel of another paper by the first two authors, where they proposed a new construction of generalized continued fractions (\(p\)-continued fraction). The main idea behind the definition is the property that its convergents are best approximations with respect to the \(L^p\) norm, for some \(p > 1\). In current paper the authors extend that construction to the region \(0 < p < 1\), where the quasinorm \(L^p\) is no longer convex. The authors provide the upper bound on the the approximation coefficients of the \(p\)-continued fraction. They prove that this estimate it is asymptotically sharp. The upper bounds on the quantity of consecutive regular convergents that are skipped by the \(p\)-continued fraction are also discussed in this paper.