5 april, Andreas Strömbergsson: On the values of a random linear form modulo one

Abstract. Given a (large) positive integer N and random numbers x,y in [0,1], let X be the number of values n=1,2,3,...,N for which the fractional part of xn+y lies in the interval [0,c/N], where c>0 is a fixed constant. It was proved by Mazel and Sinai that X has a limit distribution as N tends to infinity. This result was later extended to linear forms in several variables by Jens Marklof, who gave a geometric proof using the mixing properties of the diagonal flow on the manifold SL(n,Z)\SL(n,R). We will discuss Marklof's proof and some extensions of it, and also present results on the explicit form of the limit distribution (in the case of xn+y). We will also discuss some other problems leading to the same limit distribution.