Search Results

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 …

This thing cannot hold no matter what. As Echo rightfully commented, the expression doesn't even compile when $t=0$. It is true one might temper the integral away from $0$, but that's not what you ask …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …