Generalized Beatty sequences

A well-known result due to S. Beatty is that if α and β are positive irrational numbers satisfying α−1+β−1=1 then each positive integer is to be found in precisely one of the sequences {[kα]}, {[kβ]}(k=1,2,3,…) where [x] denotes the integral part of x. The present note generalizes this result to the case of the pair of sequences {[f(k)]}, {[g(k)]} with suitable hypotheses on the functions f and g. The special case f(x)=αx, g(x)=βx is the result due to Beatty