The α-continued fraction is a modification of the nearest integer continued fractions taking n as the integer part of y when y ε [n - 1 + α, n + α), instead of the nearest integer. For x εo[α - 1, α), we have the following α-continued fraction expansion: with Cn ≥ 1 and εn = ±1 for n ≥ 1. We prove the Borel-Bernstein theorem for α-continued fractions and also discuss some metrical properties related to max1 ≤ n ≤ N Cn. Indeed, we prove that exist and have the same constant for almost every x.
ASJC Scopus subject areas
- Algebra and Number Theory