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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?

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    Would something like this be a counter example: $k=1$, $a_{2k}=c_k$, $a_{2k-1}=-c_k$ where $c_k$ is equidistributed – Calvin Khor Dec 16 '21 at 15:25

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