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Raréfaction dans les suites b-multiplicatives

Abstract : The primary object of study is a subclass of b-multiplicative sequences, p-rarefied which means that the subsequence of terms of index multiple of a prime number p is taken. The sums of their initial terms have an asymptotic structure described by an exponent α∈]0,1[ and a contnous periodic "rarefaction function". This structure is valid for sequences with complex values in the unit disc, in both cases of the usual numerating system (section 1.1) and one with b successive digits among which there are positive and negative (section 1.2). This formalism is analogous to the formalism for the Thue-Morse sequence in texts by Gelfond; Dekking; Goldstein, Kelly, Speer; Grabner; Drmota, Skalba and others. The second, largely independent, part concerns rarefaction in sequences with terms in -1,0 or 1. Most results concern the case where b is a generator of the multiplicative group modulo p. This condition has been conjectured to be valid for infinity of primes, by Artin. The constants which are important, can be written as symmetric polynomials of P(ζ^j) where ζ is a primitive p-th root of unity, P is a polynomial with integer coefficients and j runs through the numbers from 1 to p-1 (section 1.3). The text describes a combinatorics-based method to study the values of these symmetric polynomials, where the combinatorial problem is as follows. Count the solutions of a linear congruence or a system modulo p, which satisfy a condition: the values of variables must be different from each other and from zero. Importance is attached to the difference between the numbers of solutions of two congruences that differ only in the free term. For the congruences of the form x₁+x₂+...+xₙ=i mod p this problem reduces to a well-known result. The text (section 2.2) gives an original proof of it, using the Möbius inversion formula in the p.o.set of partitions of a finite set. If at least two distinct coefficients are present, we can fix a set of coefficients (of size d) and put the answers corresponding to all possible linear congruences into an array that will be called "finite Pascal's triangle". It is a function δ:N^d→Z restricted to inputs with the sum of coordinates smaler than p (a simplex), and it has two properties. A recursive equation similar to the equation of Pascal holds everywhere except the points where the sum of coefficients is a multiple of p (a sublattice of Z^d the points of which are called "sources"); the values induced by this equation beyond the simplex are zeroes (section 2.3 and part of 2.4). An algorithm that finds the unique function delta satisfying these condiditions is described (section 2.4). It consists in successive applications of the equation in a precise order. These results are then applied to the b-multiplicative sequences (section 2.5). We also prove that the number of sources depends only on the dimention d and the size p of the simplex. We conjecture (section 2.6) that this number is the smallest possible for all numerical arrays of the same dimention and size that satisfy the same conditions. A first result about the systems of two linear congruences is proved (section 2.5.4). It is shown how these systems are related to a method by Drmota and Skalba of proving the absence of Newman's phenomenon (in a precise sence) initially described for the Thue-Morse sequence and for a prime p such that 2 is a generator of the multiplicative group modulo p, then extended to the sequence (-1)^{number of digits 2 in the ternary extension of n} called "++-". These questions generate many algorithmic and programming problems. Several sections link to illustration situated in the Annexe. Most of these figures are published for the first time.
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Submitted on : Monday, February 17, 2014 - 9:48:21 AM
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Alexandre Aksenov. Raréfaction dans les suites b-multiplicatives. Mathématiques générales [math.GM]. Université de Grenoble, 2014. Français. ⟨NNT : 2014GRENM001⟩. ⟨tel-00947586⟩



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