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A $(m,n)$ sonar sequence is an $m\times n$ array with exactly one dot in each column and where all lines connecting two dots in the array are distinct as vectors. These arrays are known to have many applications such as sonar and radar detection and these are studied as a particular case of Golomb rectangles or two-dimensional Sidon sets. The main open problem for sonar sequences is: for fixed $m$ , find the largest $n$ for which there is an $(m,n)$ sonar sequence, these sequences are called the best sonar sequences. The extended sonar sequences are generalizations of sonar sequences where each column has at most one dot, the motivation to study these arrays are the best results obtained when applied to radar and sonar detection. In this paper, we give the best sonar sequences with $m\leq 100$ obtained from an exhaustive computational search based on the Caicedo, Ruiz and Trujillo constructions and we present new constructions of extended sonar sequences that use Sidon sets.

Two new constructions for families of optical orthogonal codes are presented. The first is a generalization of the well-known construction of Sidon sets given by I. Z. Ruzsa. The second construction is optimal with respect to the Johnson bound, and its parameters (n, w, λ) are respectively (ph+1 - p, p, 1), where p is any prime, h is an integer greater than 1 and the family size is ph-1 + ph-2 + ⋯ + p2 + p.

A polynomial P ∈ 𝔽[x_1,…,x_n] is said to be symmetric if it is invariant under any permutation of its input variables. The study of symmetric polynomials is a classical topic in mathematics, specifically in algebraic combinatorics and representation theory. More recently, they have been studied in several works in computer science, especially in algebraic complexity theory. In this paper, we prove the computational hardness of one of the most basic kinds of symmetric polynomials: the monomial symmetric polynomials, which are obtained by summing all distinct permutations of a single monomial. This family of symmetric functions is a natural basis for the space of symmetric polynomials (over any field), and generalizes many well-studied families such as the elementary symmetric polynomials and the power-sum symmetric polynomials. We show that certain families of monomial symmetric polynomials are VNP-complete with respect to oracle reductions. This stands in stark contrast to the case of elementary and power symmetric polynomials, both of which have constant-depth circuits of polynomial size., LIPIcs, Vol. 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022), pages 16:1-16:14

Let \(S \subseteq \mathbb N\) be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let \(L=S \cap [0,c-1]\) and \(W(S)=|P||L|-c\). In 1978, Herbert Wilf asked whether \(W(S) \ge 0\) always holds, a question known as Wilf’s conjecture and open since then. A related number \(W_0(S)\), satisfying \(W_0(S) \le W(S)\), has recently been introduced. We say that S is a near-miss in Wilf’s conjecture if \(W_0(S)

This thesis studies problems in extremal graph theory, combinatorial number theory, and finite incidence geometry, and the interplay between these three areas. The first topic is the study of the Tur\'an number for $C_4$. F\"uredi showed that $C_4$-free graphs with $\mathrm{ex}(n, C_4)$ edges are intimately related to polarity graphs of projective planes. We prove a general theorem about dense subgraphs in a wide class of polarity graphs, and as a result give the best-known lower bounds for $\mathrm{ex}(n, C_4)$ for many values of $n$. We also study the chromatic and independence numbers of polarity graphs, with special emphasis on the graph $ER_q$. Next we study Sidon sets on graphs by considering what sets of integers may look like when certain pairs of them are restricted from having the same product. Other generalizations of Sidon sets are considered as well.We then use $C_4$-free graphs to prove theorems related to solvability of equations. Given an algebraic structure $R$ and a subset $A\subset R$, define the {\em sum set} and {\em product set} of $A$ to be $A+A = \{a+b:a,b\in A\}$ and $A\cdot A = \{a\cdot b: a,b\in A\}$ respectively. Showing under what conditions at least one of $|A+A|$ or $|A\cdot A|$ is large has a long history of study that continues to the present day. Using spectral properties of the bipartite incidence graph of a projective plane, we deduce that nontrivial sum-product estimates hold in the setting where $R$ is a finite quasifield. Several related results are obtained.Finally, we consider a classical question in finite incidence geometry: what is the subplane structure of a projective plane? A conjecture widely attributed to Neumann is that all non-Desarguesian projective planes contain a Fano subplane. By studying the structural properties of polarity graphs of a projective plane, we show that any plane of even order $n$ which admits a polarity such that the corresponding polarity graph has exactly $n+1$ loops must contain a Fano subplane. The number of planes of order up to $n$ which our theorem applies to is not bounded above by any polynomial in $n$.

Le document contient 52 pages et 1 figures. C'est une nouvelle version d'un papier de l'auteur déposé le 26 Août 2015 sur Arxiv. Mais cette nouvelle version contient une nouvelle démonstration qui utilise des résultats fins de la théorie d'interpolation de Carleson. Quelques détails sont ajoutés et les grandes lignes de la preuve exposées.; Using the combinatorial Singer's construction, we exhibit a sequence of flat polynomials with coefficients $0,1$. We thus get that there exist a sequences of Newman polynomials that are $L^\alpha$-flat, $0 \leq \alpha

We construct several families of perfect sublattices with minimum $4$ of $\mathbb Z^d$. In particular, the number of $d-$dimensional perfect integral lattices with minimum $4$ grows faster than $d^k$ for every exponent $k$., Final version, to appear in Journal de Th{\'e}orie des Nombres de Bordeaux

We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s_1+s_2, s_i\in S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g\ge 3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Koulantzakis' idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo & Ruzsa & Trujillo, Jia, and Habsieger & Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases., 15 pages, 1 figure (revision fixes typos, adds a few details, and adjusts notation)

Let h ≥ 2 be an integer. We say that a set A of positive integers is an asymptotic basis of order h if every large enough positive integer can be represented as the sum of h terms from A. A set of positive integers A is called a Sidon set if all the sums a+b with a,b ∈ A, a ≤ b are distinct. In this paper we prove the existence of Sidon set A which is an asymptotic basis of order 4 by using probabilistic methods. 2000 AMS Mathematics subject classification number: primary: 11B13, secondary: 11B75.

International audience; We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space $c_0$.

We study s(k,n), the maximum size of A+A where A is a k-subset of [n]. A few known functions from additive number theory can be expressed via s(k,n). For example, our estimates of s(k,n) imply new bounds on the maximum size of quasi-Sidon sets, a problem posed by Erdos and Freud [J. Number Th.38 (1991) 196-205]. Also, applications to so-called edge-magic labellings of graphs are given., 19 pages; 4 figures

We investigate translation invariant subspaces of the space of uniformly convergent Fourier series and Orlicz spaces, with finite cotype. In the case of Orlicz spaces, this leads to some new characterizations of p -Rider sets.