We are interested in permutations preserving certain distribution properties of sequences. In particular we consider $\mu$-uniformly distributed sequences on a compact metric space $X$, 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of $Aut\left(\mathbf{N}\right)$ leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group $𝒢$. We show that $𝒢$ is big in the...

Let $S$ be an abelian semigroup, and $A$ a finite subset of $S$. The sumset $hA$ consists of all sums of $h$ elements of $A$, with repetitions allowed. Let $\left|hA\right|$ denote the cardinality of $hA$. Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial $p\left(t\right)$ such that $\left|hA\right|=p\left(h\right)$ for all sufficiently large $h$. Lattice point counting is also used to prove that sumsets of the form $h_1{A}_1+\cdots +h_r{A}_r$ have multivariate polynomial growth.

For any partition of a set of squarefree numbers with relative density greater than 3/4 into two parts, at least one part contains three numbers whose product is a square. Also generalizations to partitions into more than two parts are discussed.