### 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.