The semigroup of nonempty finite subsets of rationals.
The sequence of fractional parts of roots
We study the function , where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of , that if log θ is rational, then for all but finitely many positive integers n, . We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy . Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued...
The set of minimal distances in Krull monoids
Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say . The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds if every...
The set of rational cycles for the 3x+1 problem
The size of sums of sets
The space of period polynomials
The square-free kernel of
The structure of maximal zero-sum free sequences
The structure of maximum subsets of with no solutions to .
The structure of second order sequences in a finite field.
The terms in Lucas sequences: an algorithm and applications to diophantine equations
The terms in Lucas sequences divisible by their indices.
The terms of the form 7kx² in the generalized Lucas sequence with parameters P and Q
Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.
The topological structure of the set of subsums of an infinite series
The Tribonacci substitution.
The wildcard set's size in the density Hales-Jewett theorem.
The zero-multiplicity of ternary recurrences
Théorème des nombres premiers pour les fonctions digitales
The aim of this work is to estimate exponential sums of the form , where Λ denotes von Mangoldt’s function, f a digital function, and β ∈ ℝ a parameter. This result can be interpreted as a Prime Number Theorem for rotations (i.e. a Vinogradov type theorem) twisted by digital functions.
Theoretical and computational bounds for m-cycles of the 3n+1-problem
Théorie additive des nombres. Densité de Schnirelman