Abstract. We answer a question of Erdős by showing that the least modulus of a distinct covering system is at most 10 16 .

Abstract. We study the random walk on the symmetric group S n generated by the conjugacy class of cycles of length k. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after n k log n steps, uniformly in k = o(n) as n → ∞. The analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles.

We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice Z 2 . We also determine the asymptotic spectral gap, asymptotic mixing time and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus pZ{mZq 2 . The techniques use analysis of the space of functions on Z 2 which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in ℓ p pZ 2 q as linear combinations of certain discrete derivatives of Green's functions, extending a result of Schmidt and Verbitskiy [35].2010 Mathematics Subject Classification. Primary 82C20, 60B15, 60J10. Key words and phrases. Abelian sandpile model, random walk on a group, spectral gap, cutoff phenomenon, critical density, harmonic modulo 1 functions.

Given a ±1 random completely multiplicative function f, we prove by estimating moments that the limiting distribution of the normalized sum converges to the standard Gaussian distribution as x → ∞ when r restricts summation to n having o(log log log x) prime factors. We also give an upper bound for the large deviations of with the sum restricted to numbers having a fixed number k of prime factors.

Abstract. We consider an analogue of the Kac random walk on the special orthogonal group SO(N ), in which at each step a random rotation is performed in a randomly chosen 2-plane of R N . We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large N limit. In the special case where the angle of rotation is deterministic this confirms a conjecture of Rosenthal [19]. Under mild conditions we also establish a cut-off for convergence of the walk to stationarity under the L 2 norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and L 2 norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis.