The distribution of sequences in residue classes
The distribution of sequences in residue classes
We prove that any set of integers ${\mathcal A}\subset [1,x]$ with $\vert {\mathcal A} \vert \gg (\log x)^r$ lies in at least $\nu _{\mathcal A}(p) \gg p^{\frac {r}{r+1}}$ many residue classes modulo most primes $p \ll (\log x)^{r+1}$. (Here $r$ is a positive constant.) This generalizes a result of ErdÅs …