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This question has been considered by Lemanczyk and others, and Lemanczyk developed a quite general way to produce dynamical systems which are not disjoint from Mobius (unfortunately or fortunately, de …

In the London proceedings (Cassels-Froehlich), Tate and Serre have written some (classical) exercises regarding CFT (i.e. deducing higher reciprocity laws from Artin's reciprocity law, the Hasse-Minko …

This is a very interesting question, which actually asks about the interplay between equidistribution (or harmonic analysis if you would like to call it that way) and ergodic theory. As Vaughn mentio …

A very readable introduction is Kurlberg's paper - http://www.math.kth.se/~kurlberg/eprints/short_expsum.pdf

Take a look at Cassels - "An intorduction to diophantine approximation", Theorem VI in Ch1, where the theorem that Gerry mentioned is proved. I'm guessing that it appears also in Siegel's book about t …

For getting the every-point statement, at-least in the compact abelian case (see Tao's comment above), one can either prove it by harmonic analytic approach (Weyl's equi. criterion + van der corput tr …

There are two ways to solve this problem - one by ergodic methods, and the other one using purely harmonic methods. The harmonic method you are indicating is just to take the delta function of the po …

This is not exactly what you've asked for, but I'll address this article directly, because it is not related to automorphic L-functions "directly" but more to homogeneous dynamics. You can actually re …

A very thorough (although a bit dense) modern treatment is given in the article of Einsiedler, Lindenstrauss, Michel and Venkatesh about the ergodic theoretical proof of Duke's theorem (which is not u …

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 …

I'm pretty sure that plenty of those kind of questions are covered in Cassels' book. The modern approach to this kind of problems follows from dynamics on homogeneous spaces via Dani's correspondence …

While Peter Humphries' answer is entirely correct for the question asked by the OP, the technique indicated there is far from addressing the most general situation. The most basic technique towards t …

It is a whole line of ideas (and proofs) which go usually by the name of ``Linnik's problems''. Apart from Linnik's book (and the Linnik-Skubenko theorem), it has been extensively studied by many rese …