## EVENT DETAILS AND ABSTRACT

**Number Theory**
**Title:** The Kesten problem in higher dimensions

**Speaker:** Matthew Welsh

**Speaker Info:** UMD

**Brief Description:**

**Special Note**:

**Abstract:**

For real $\alpha$ 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 $I$ with $n \leq N$ 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.

**Date:** Friday, March 15, 2024

**Time:** 2:00PM

**Where:** Lunt 103

**Contact Person:** Maksym Radziwill

**Contact email:** maksym.radziwill@gmail.com

**Contact Phone:**

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