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We are interested in permutations preserving certain distribution properties of sequences. In particular we consider -uniformly distributed sequences on a compact metric space , 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group . We show that is big in the...
Let be an abelian semigroup, and a finite subset of . The sumset consists of all sums of elements of , with repetitions allowed. Let denote the cardinality of . Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial such that for all sufficiently large . Lattice point counting is also used to prove that sumsets of the form 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.
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