 Beatty sequence reciprocity

Let $\alpha >1$ be irrational and $\beta =\alpha /(\alpha -1)$, so $1/\alpha +1/\beta =1$. The Beatty sequences $\lfloor n\alpha \rfloor$ and $\lfloor n\beta \rfloor$, $n\ge 1$, partition the positive integers.

Indeed, the numbers of terms at most $m$ are $\lfloor (m+1)/\alpha \rfloor$ and $\lfloor (m+1)/\beta \rfloor$. Their arguments sum to the integer $m+1$, and neither is integral, so the counts sum to $m$. Subtracting the count at $m-1$ shows that every integer occurs exactly once in the two lists together. This proof interprets reciprocity as an exact complementarity of two densities.