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This sum is explicitly studied in the recent paper by Anderson-Cook-Hughes-Kumchev https://arxiv.org/abs/1703.02713 , they refer to a an estimate of Shparlinski obtaining a bound of $O(q^{1/2+\epsilon …

Let $S$ be the semi-group generated by $2$ and $3$. The interest in this semi-group is that it is non-lacunary, meaning $s_{n+1}/s_{n} \rightarrow 1$. A famous theorem due to Furstenberg will tell yo …

Basically you don't need the Weyl's Equi. theorem, it's enough to use Kronecker's lemma about density. If you want to use measure theory, then your question follows from any ergodic theorem you would …

It is unclear what is "most $r$'s even mean. A standard argument would show that for any increasing sequence, for Lebesgue almost every $x$, $a_{n}.x$ is equidistributed mod $1$. For the case of power …

There is some confusion here, as literally the construction of complementary series in $SL_{2}$ will give you unitary representations with arbitrary slow decay. For any homogeneous space $G/\Gamma$, t …