For α > 1, set β = 1/(α − 1). We show that, for every 1 < α < (, where ζ denotes the Riemann zeta function. We use this result to derive an asymptotic formula for the number of triplets (l, m, n) of positive integers such that l < x and ⌊l α ⌋ + ⌊m α ⌋ = ⌊n α ⌋.