【摘要】
For an irrational $\alpha$ and an interval $ I = [x, x +u] \subset \RR/\ZZ$, consider the discrepancy $D_N(\alpha, x, u)$ between the number of $n \leq N$ with $n \alpha \pmod 1 $ in $I$ and the expected number, $uN$. In the 1960s, Kesten proved that if $\alpha$ and $x$ are taken uniformly at random, then $D_N$ normalized by $\rho(u) \log N$ converges to a Cauchy distribution. Similar results are known in higher dimensions due to work by Dolgopyat and Fayad; however with extra randomness akin to taking $u$ random as well. In work in progress with Dolgopyat and Fayad, we succeed in removing this extra randomness by proving extensions of equidistribution results on $\SL(d, \RR ) \ltimes \RR^d$ modulo $\SL(d, \ZZ) \ltimes \ZZ^d$ due to Strombergsson and Kim.
