Does equidistribution of $(a_n)_{n\in\mathbb{N}}$ imply equidistribution of $(a_n+a_{n+k})_{n\in\mathbb{N}}$?

Question

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers that is equidistributed modulo 1 and let $k\in\mathbb{N}$.

Then it is clear that the sequence $(a_{n+k})_{n\in\mathbb{N}}$ is also equidistributed modulo 1. However, if $b_n=a_n+a_{n+k}$, for $n\in\mathbb{N}$, is it true that $(b_n)_{n\in\mathbb{N}}$ is also equidistributed?