LARGE FAMILIES OF PSEUDORANDOM SUBSETS FORMED BY POWER RESIDUES

International audienceIn an earlier paper the authors introduced the measures of pseudo-randomness of subsets of the set of the positive integers not exceeding N , and they also presented two examples for subsets possessing strong pseudorandom properties. One of these examples included permutation polynomials f (X) ∈ F p [X] and d-powers in F p. This construction is not of much practical use since very little is known on permutation polynomials and there are only very few of them. Here the construction is extended to a large class of polynomials which can be constructed easily, and it is shown that all the subsets belonging to the large family of subsets obtained in this way possess strong pseudorandom properties. The complexity of this large family is also studied

Exponential sums with reducible polynomials

International audienceHooley proved that if f ∈ Z[X] is irreducible of degree ≥ 2, then the fractions {r/n}, 0 < r < n with f (r) ≡ 0 (mod n), are uniformly distributed in ]0, 1[. In this paper we study such problems for reducible polynomials of degree 2 and 3 and for finite products of linear factors. In particular, we establish asymptotic formulas for exponential sums over these normalized roots

Structure et enregistrement des traces latentes d'ions argon et fer dans l'olivine et le mica muscovite

On étudie, par diffusion des rayons X aux petits angles, la nature de la trace latente des ions Fe et Ar (énergie comprise entre 1 et 7 MeV/nucléon) dans l'olivine et le mica muscovite. Dans les deux cas, il se forme des zones endommagées qu'on caractérise par leur taille et leur espacement. La distance inter-zones conditionne la mise en évidence des traces par attaque chimique : l'enregistrement des traces n'apparaît pas lié à un mécanisme à seuil énergétique

Arithmetic Properties of Summands of Partitions

International audienceLet $d\ge 2$ be an integer. We prove that for almost all partitions of an integer the parts are well distributed in residue classes mod d. The limitations of the uniformity of this distribution are also studied

Sommes des chiffres de multiples d'entiers

52 pagesLet q in N, q >= 2. For n in N, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. We give upper bounds for exponential sums like G(x,y, theta ; a,h) = sum_{x= 2 by Coquet and Solinas. For r >= 2, our results are more precises and significative for a wider range of h. Furthermore they are uniform in x and theta and explicits in h. The control of these parameters is crucial for various applications given in the paper. For exemple we prove that if k in N, k >=2, there exists infinitely many integers n with exactly k prime factors and such that s_q(n) md am (for (m,q-1)=1). We also obtain upper bounds of sums of the form sum_{n le x} exp (2i pi alpha s_q(hn))f(n) where f is a multiplicative fonction of modulus less than 1

Sums of proper divisors with missing digits

Let $s(n)$ denote the sum of proper divisors of an integer $n$. In 1992, Erd\H{o}s, Granville, Pomerance, and Spiro (EGPS) conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then $s^{-1}(\mathcal{A})$ also has asymptotic density zero. In this paper we show that the EGPS conjecture holds when $\mathcal{A}$ is taken to be a set of integers with missing digits. In particular, we give a sharp upper bound for the size of this preimage set. We also provide an overview of progress towards the EGPS conjecture and survey recent work on sets of integers with missing digits

Local distribution of the parts of unequal partitions in arithmetic progressions I

International audienceIn [4], András Sárközy and the authors proved that for almost all unequal partitions of an integer n, the parts are evenly distributed in residue classes modulo d for d = o(n^(1/2)). In this paper, we study very precisely the local distribution in arithmetic progressions of the parts of unequal partitions. We obtain some asymptotic formulae for the number of unequal partitions of n with exactly N r parts congruent to r mod d, 1 6 r 6 d. Our results show that (N 1 ,. .. , N d) behaves like a random Gaussian vector. This illustrates the fact that the distribution of the parts of unequal partitions in residue classes is much more uniform that in the case of unrestricted partitions