On Integers n Relatively Prime to [αn]

Abstract

The object of this paper is to consider a problem suggested by Dr. K. F. Roth, on the distribution of integers n that are relatively prime to the integral part of an, a being a fixed real number. He conjectured that the number of positive integers up to x with this property is asymptotic to (or in other words that they have the density ), for irrational a. I prove this and rather more in the following