Some metric properties of α-continued fractions Original Research Article

The α-continued fraction is a modification of the nearest integer continued fractions taking n as the integer part of y when yset membership, variant[n−1+α,n+α), instead of the nearest integer. For xset membership, variant[α−1,α), we have the following α-continued fraction expansion:imagewith cngreater-or-equal, slanted1 and var epsilonn=±1 for ngreater-or-equal, slanted1. We prove the Borel–Bernstein theorem for α-continued fractions and also discuss some metrical properties related to max1less-than-or-equals, slantnless-than-or-equals, slantN cn. Indeed, we prove thatimageexist and have the same constant for almost every x.