Additive Energy and Irregularities of Distribution

We consider strictly increasing sequences (an)n≥1 of integers and sequences of fractional parts ({anα})n≥1 where α ∈ R. We show that a small additive energy of (an)n≥1 implies that for almost all α the sequence ({anα})n≥1 has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.

When does a semiring become a residuated lattice?

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.

A primality criterion based on Lucas’ congruence

Let [Formula: see text] be a prime. Back in 1878, Lucas proved that the congruence [Formula: see text] holds for any nonnegative integer [Formula: see text]. The converse statement was given in Problem 1494 of Mathematics Magazine proposed in 1997 by Deutsch and Gessel. In this note, we generalize this converse assertion as follows: If [Formula: see text] and [Formula: see text] are integers such that [Formula: see text] for every integer [Formula: see text], then [Formula: see text] is a prime and [Formula: see text] is a power of [Formula: see text].

On a relation between sums of arithmetical functions and Dirichlet series

We introduce a concept called good oscillation. A function is called good oscillation, if its m-tuple integrals are bounded by functions having mild orders. We prove that if the error terms coming from summatory functions of arithmetical functions are good oscillation, then the Dirichlet series associated with those arithmetical functions can be continued analytically over the whole plane. We also study a sort of converse assertion that if the Dirichlet series are continued analytically over the whole plane and satisfy a certain additional assumption, then the error terms coming from the summatory functions of Dirichlet coefficients are good oscillation.

Reversibility and Acyclicity

It is well-known that the transition matrix of a reversible Markov process can have only real eigenvalues. An example is constructed which shows that the converse assertion does not hold. A generalised notion of reversibility is proposed, ‘dynamic reversibility’, which has many of the implications for the form of the transition matrix of the classical definition, but which does not exclude ‘circulation in state-space’ or, indeed, periodicity.