On the Sz\"usz's Solution to Gauss' Problem

TL;DR

This paper offers a new proof of Sz"usz's solution to Gauss' problem on continued fractions, improving the value of a key parameter using properties of the Perron-Frobenius operator and Gauss' measure.

Contribution

It provides a novel proof of Sz"usz's theorem on Gauss' problem, optimizing the value of q with a different approach involving the Perron-Frobenius operator.

Findings

Derived the value 0.7594 for q, an improvement over Sz"usz's 0.485.

Utilized properties of the Perron-Frobenius operator of the continued fraction transformation.

Presented a new proof technique for a classical problem in continued fractions.

Abstract

The present paper deals with Gauss' problem on continued fractions. We present a new proof of a theorem which Sz\"usz applied in order to solve this problem. To be noted, that we obtain the value $0.7594...$ for $q$, which has been optimized by Sz\"usz in his 1961 paper "\"Uber einen Kusminschen Satz", where the value 0.485 is obtained for $q$. In our proof, we make use of an important property of the Perron-Frobenius operator of $\tau$ under $\gamma$, where $\tau$ is the continued fraction transformation, and $\gamma$ is the Gauss' measure.