Representing real numbers in Möbius number systems
Rep-tiling Euclidean space.
Search for Wieferich Primes through the use of Periodic Binary Strings
The result of the distributed computing projectWieferich@Home is presented: the binary periodic numbers of bit pseudo-length j ≤ 3500 obtained by replication of a bit string of bit pseudo-length k ≤ 24 and increased by one are Wieferich primes only for the cases of 1092 or 3510.
Searching for Kaprekar's constants: algorithms and results.
Semihappy numbers.
Sequences of generalized happy numbers with small bases.
Séries formelles et théorie des automates.
Sets of -expansions and the Hausdorff measure of slices through fractals
We study natural measures on sets of -expansions and on slices through self similar sets. In the setting of -expansions, these allow us to better understand the measure of maximal entropy for the random -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading...
Signed bits and fast exponentiation
An exact analysis is given of the benefits of using the non-adjacent form representation for integers (rather than the binary representation), when computing powers of elements in a group in which inverting is easy. By counting the number of multiplications for a random exponent requiring a given number of bits in its binary representation, we arrive at a precise version of the known asymptotic result that on average one in three signed bits in the non-adjacent form is non-zero. This shows that...
Some binary representations of real numbers and odd solutions of the functional equation ... .
Some euclidean properties for real quadratic fields
Some further results concerning (j,ε) -normality in the rationals
Some number-theoretical properties of generalized sum-of-digit functions
Sommes des chiffres de multiples d'entiers
Soit , . Pour , on note la somme des chiffres de en base . Nous donnons des majorations de sommes d’exponentielles de la formepour , et . De telles sommes ont déjà été étudiées dans le cas par Gelfond, et pour entre autre par Coquet et Solinas. Nos résultats étendent le domaine de validité en de ces précédents travaux pour , sont plus précis et ont l’avantage d’être uniformes en et et effectifs en . Ce contrôle soigneux des paramètres nous permet d’obtenir divers types d’applications....
Sommes des chiffres et nombres presque premiers.
Speeding up the computations on an elliptic curve using addition-subtraction chains
Strongly automatic semigroups
Dans cet article, nous introduisons la notion de semi-groupe fortement automatique, qui entraîne la notion d’automaticité des semi-groupes usuelle. On s’intéresse particulièrement aux semi-groupes de développements en base , pour lesquels on obtient un critère de forte automaticité.
Structure of the Markoff spectrum below √12
Substitutions, abstract number systems and the space filling property
In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.
Sums of digits, overlaps, and palindromes.