arXiv subjects

Carlos Vinuesa

Publications and source records attributed to Carlos Vinuesa.

4 recordsLinked to original sources

arXiv2012

Asymptotics for Magic Squares of Primes

Based on the work of Green, Tao and Ziegler, we give asymptotics when $N \to \infty$ for the number of $n \times n$ magic squares with their entries being prime numbers in $[0,N]$. For every $n \ge 3$ we give appropriate systems of linear forms (or equivalently basis) describing all $n \times n$ magic squares with integer entries and we calculate the complexity of these systems in the Green and Tao sense. We compute the precise asymptotics for the cases $n=3$ (complexity 3) and $n=4$ (complexity 1), and the given algorithm works for $n \ge 5$ (complexity 1). Finally, we show that the asymptotics are exactly the same if we impose that all the entries of the magic squares have to be different.

math.NT↗

arXiv2009

Generalization of a theorem of Erdos and Renyi on Sidon Sequences

Erd\H os and Rényi claimed and Vu proved that for all $h \ge 2$ and for all $ε> 0$, there exists $g = g_h(ε)$ and a sequence of integers $A$ such that the number of ordered representations of any number as a sum of $h$ elements of $A$ is bounded by $g$, and such that $|A \cap [1,x]| \gg x^{1/h - ε}$. We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a $g$ that satisfies $g_h(ε) \ll ε^{-1}$, improving the bound $g_h(ε) \ll ε^{-h+1}$ obtained by Vu. Finally we use the "alteration method" to get a better bound for $g_3(ε)$, obtaining a more precise estimate for the growth of $B_3[g]$ sequences.

math.NT↗

arXiv2009

Generalized Sidon sets

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.

math.NT↗

arXiv2009

Improved bounds on the supremum of autoconvolutions

We give a slight improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we disprove a long standing natural conjecture of Schinzel and Schmidt concerning the extremal function for such autoconvolutions.

math.CA↗