A sequence of integers $(x_n)$ is equidistributed mod $p$ if for all $a$ mod $p$, we have as $n \to \infty$: $$ \dfrac{1}{X} \# \{n < X: x_n \equiv a \mod p \} \to \dfrac{1}{p}.$$

Let $(a_n)$ and $(b_n)$ be two integer sequences. Define the convolution of $(a_n)$ and $(b_n)$ to be the sequence $(a_n \star b_n)$, where: $$ a_n \star b_n = \sum_{k=0}^n a_k b_{n-k}.$$

What are sufficient conditions we can impose on the sequences $(a_n)$ and $(b_n)$ that ensure that $(a_n \star b_n)$ is equidistributed mod $p$?



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