Javier Cilleruelo

• Autonomous University of Madrid

Combining previous ideas from Garaev and the first author, we prove a general theorem to estimate the number of elements of a subset A of an abelian group G=Zn1×⋯×Znk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setle...

In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erd\"os and R\'enyi in 1960. In 1975, Goguel proved that such a sequence is almost surely not an asymptotic basis of order s. Our first result asserts that it is however almost surely a basis of order s + x for any x > 0. We then study the s-fold sumset sA = A + ......

In this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and | ${\mathcal U}$ |, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having s...

We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$, respectively) and starting from the origin. We show that, almost surely, the asymptotic proportion of strings of $k$ consecutive visible lattice points visited by such an $\alpha$-random walk is...

Erdos conjectured the existence of an infinite Sidon sequence of positive integers which is an asymptotic basis of order 3. We progress towards this conjecture in several directions. We prove the conjecture for all cyclic groups ℤN with N large enough. We also show that there is an infinite B2[2] sequence which is an asymptotic basis of order 3. Fi...

We study th etypical behavior of the size of the ratio set A/A for a random subset A in {1, . . . , n}.

We study the probability that a cycle of length k in the lattice [1, n]^s does not contain more lattice points than the k vertices of the cycle. Then we introduce the chromatic zeta fuction of a graph to generalize this problem to other configurations induced by a given graph H.

We study the probability that a cycle of length k in the lattice [1, n]^s does not contain more lattice points than the k vertices of the cycle. Then we generalize this problem to other con?gurations induced by a given graph H introducting the chromatic zeta fuction of a graph.

We prove that for each odd number k, the sequence (k2(n) + 1)(n >= 1) contains only a finite number of Carmichael numbers. We also prove that k = 27 is the smallest value for which such a sequence contains some Carmichael number.

We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs.

In the present paper we obtain several new results related to the problem of upper bound estimates for the number of solutions of the congruence $$ x^{x}\equiv \lambda\pmod p;\quad x\in \mathbb{N},\quad x\le p-1, $$ where $p$ is a large prime number, $\lambda$ is an integer corpime to $p$. Our arguments are based on recent estimates of trigonometri...

For h = 3 and h = 4 we prove the existence of infinite Bh sequences B with counting function [equation presented] This result extends a construction of I. Ruzsa for B2 sequences.

In his famous 1946 paper, Erdős (1946) proved that the points of a n×n portion of the integer lattice determine Θ(n/logn) distinct distances, and a variant of his technique derives the same bound for n×n portions of several other types of lattices (e.g., see Sheffer (2014)). In this note we consider distinct distances in rectangular lattices of the...

We use elementary methods to prove an incidence theorem for points and spheres in $\mathbb{F}_q^n$. As an application, we show that any point set of $P\subset \mathbb{F}_q^2$ with $|P|\geq 53q^{4/3}$ determines a positive proportion of all circles. The latter result is an analogue of Beck's Theorem for circles.

For a given finite field \(\mathbb F _q\), we study sufficient conditions to guarantee that the set \(\{\theta _1^x+\theta _2^y:\ 1\le x\le M_1,\ 1\le y\le M_2\}\) represents all the nonzero elements of \(\mathbb F _q\). We investigate the same problem for \(\theta _1^x-\theta _2^y\) and as a consequence we prove that any element in the finite fiel...

Bourgain, Konyagin and Shparlinski obtained a lower bound for the size of the product set AB when A and B are sets of positive rational numbers with the numerator and denominator less than or equal to Q. We extend and slightly improve that lower bound using a different approach.

We study the length of the gaps between consecutive members in the sumset sA when A is a pseudo s-th power sequence, with s>1. We show that, almost surely, limsup (b_{n+1}-b_{n})/log (b_n) = s^s s!/\Gamma^s(1/s), where b_n are the elements of sA.

We obtain a sharp upper bound estimate of the form $Hp^{o(1)}$ for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this bo...

We construct an infinite graph G that does not contain cycles of length four having the property that the sequence of graphs G_n induced by the first n vertices has minimum degree delta(G_n)> n^{sqrt(2)-1+o(1)}.

We construct a set of positive integers A in {1,..., n} with |A|>> n^{2/3} that does not contain Hilbert cubes of dimension 3. As a consequence we prove that ex(n; K^(3)(2,2,2))>> n^{8/3} where K^(3)(2,2,2) is the simplest complete 3-partite hypergraph. This is the first case of an improvement on the trivial lower bound for ex(n; L) when L is a com...

We construct a set of positive integers A in {1,.., n} with |A|>> n^{2/3} that does not contain Hilbert cubes of dimension 3.

Let b ≥ 2 be a fixed positive integer. We show for a wide variety of sequences {a n } n=1∞ that for almost all n the sum of digits of a n in base b is at least c b log n, where c b is a constant depending on b and on the sequence. Our approach covers several integer sequences arising from number theory and combinatorics.

We prove that for any base $b\ge 2$ and for any linear homogeneous recurrence sequence $\{a_n\}_{n\ge 1}$ satisfying certain conditions, there exits a positive constant $c>0$ such that $\# \{n\le x:\ a_n \;\text{ is} \text{ palindromic} \text{ in} \text{ base}\; b\} \ll x^{1-c}$ .

We obtain analogues of several recent bounds on the number of solutions of polynomial congruences modulo a prime with variables in short intervals in the case of polynomial equations in high degree extensions of finite fields. In these settings low-dimensional affine spaces play the role of short intervals and thus several new ideas are required.

In this note we consider distinct distances determined by points in an integer lattice. We first consider Erdos's lower bound for the square lattice, recast in the setup of the so-called Elekes-Sharir framework \cite{ES11,GK11}, and show that, without a major change, this framework \emph{cannot} lead to Erdos's conjectured lower bound. This shows t...

Erd\"os conjectured the existence of an infinite Sidon sequence of positive integers which is also an asymptotic basis of order 3. We make progress towards this conjecture in several directions. First we prove the conjecture for all cyclic groups Z_N with N large enough. In second place we prove by probabilistic methods that there is an infinite B_...

We give various results about the distribution of the sequence {a n/n}n ≥ 1 modulo 1, where a ≥ 2 is a fixed integer. In particular, we find an explicit infinite subsequence A such that {a n/n}n∈A is uniformly distributed modulo 1. Also we show that for any constant c > 0 and a sufficiently large N, the fractional parts of the first N terms of this...

We give an explicit construction of an infinite Sidon sequence A of positive integers with counting function A(x) = x root(2-1+o(1)). Ruzsa proved the existence of a Sidon sequence with similar counting function but his proof was not constructive. Our method generalizes to Bh sequences when h >= 3. In this case our constructions are not explicit bu...

For $h=3$ and $h=4$ we prove the existence of infinite $B_h$ sequences $\B$ with counting function $$\mathcal{B}(x)= x^{\sqrt{(h-1)^2+1}-(h-1) + o(1)}.$$ This result extends a construction of I. Ruzsa for $B_2$ sequences.

Foraprime p and a given square box, B, we consider all elliptic curves E r,s: Y 2 = X 3 +rX +s defined over a field F pp of p elements with coefficients (r, s) ∈ B. We obtain a nontrivial upper bound for the number of such curves which are isomorphic to a given one over F p, in terms of the size of B. We also give an optimal lower bound on the numb...

We prove that $\log lcm\{a\in A\}=n\log 2+o(n)$ for almost every set $A\subset \{1,..., n\}$. We also study the typical behavior of the logarithm of the least common multiple of sets of integers in $\{1,..., n\}$ with prescribed size. For example, we prove that, for any $0<\theta<1$, $\log lcm\{a\in A\}=(1-\theta)n^{\theta}\log n +o(n^{\theta})$ fo...

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x) \equiv y \pmod p$ and $f(x) \equiv y^2 \pmod p,$ with a prime $p$ and a polynomial $f$, where $(x,y)$ belongs to an arbitrary square with side length $M$. We use these results and methods to derive non-trivial upper bounds for the number of h...

For a polynomial f ∈ F p [X], we obtain an upper bound on the number of points (x, f (x)) modulo a prime p which belong to an arbi-trary square with the side length H. Our results is based on the Vino-gradov mean value theorem. Using these estimates we obtain results on the expansion of orbits in dynamical systems generated by nonlin-ear polynomial...

A Lehmer number is a composite positive integer n such that ϕ(n)|n − 1. In this paper, we show that given a positive integer g > 1 there are at most finitely many Lehmer numbers which are repunits in base g and they are all effectively computable. Our method is effective and we illustrate it by showing that there is no such Lehmer number when g ∈ [...

For a prime $p$ and a polynomial $f \in \F_p[X]$, we obtain upper bounds on the number of solutions of the congruences $f(x) \equiv y \pmod p \mand f(x) \equiv y^2 \pmod p,$ where $(x,y)$ belongs to an arbitrary square with side length $M$. Further, we obtain non-trivial upper bounds for the number of hyperelliptic curves $Y^2=X^{2g+1} + a_{2g-1}X^...

In this note, we give a lower bound for the distance between the maximal and minimal element in a multiplicative magic square of dimension r whose entries are distinct positive integers.

Erdős and Rényi claimed and Vu proved that for all h ≥ 2 and for all ϵ > 0, there exists g = gh(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 ∩ [1,x]| ≫ x1/h-ϵ. We give two new proofs of this result. The first one consists of an explicit co...

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b 1 b 2 a i a j with a i , a j ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps b n+1 − b n ≪ α −3 and that for t ≥ 2 there are infinitely many t-gaps b n+t − b n ≪ t 2 α −4. Furthermore, we prove that these est...

We study finite and infinite Sidon sets in ℕ d . The additive energy of two sets is used to obtain new upper bounds for the cardinalities of finite Sidon subsets of some sets as well as to provide short proofs of already known results. We also disprove a conjecture of Lindström on the largest cardinality of a Sidon set in [1,N]×[1,N] and relate it...

We give a necessary and sufficient condition on a given family A\mathcal{A} of finite subsets of integers for the Cauchy–Davenport inequality |A + B| ³ |A|+ |B|- 1,\vert \mathcal{A} + \mathcal{B}\vert \geq \vert \mathcal{A}\vert + \vert \mathcal{B}\vert - 1, to hold for any family B\mathcal{B} of finite subsets of integers. We also describe th...

Odd indexed Fibonacci numbers F2n+1 can be written as sums of two squares a 2+b 2. In this paper, we study the distribution of the lattice points (a,b) on the circles of radiusÖ{F2n+1}\sqrt{F_{2n}+1}. KeywordsFibonacci numbers-Sums of two squares Mathematics Subject Classification (2000)11B39-11P21

Let $p$ be a large prime number, $K,L,M,\lambda$ be integers with $1\le M\le p$ and ${\color{red}\gcd}(\lambda,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M; K,L)$ of solutions of the congruence $$ xy\equiv\lambda \pmod p, \qquad K+1\le x\le K+M,\quad L+1\le y\le L+M $$ and for the number $I_3(M;L)$ of s...

We use the probabilistic method to prove that for any positive integer g there exists an infinite B2[g] sequence A= {ak} such that ak ≤ k2+1/g(log k)1/g+o(1) as k→∞. The exponent 2+1/g improves the previous one, 2 + 2/g, obtained by Erdo{double acute}s and Renyi in 1960. We obtain a similar result for B2[g] sequences of squares. © 2010 Institute of...

We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solubility of some equations and distribution of sequences in small intervals. We obtain classic and more recent results avoiding the use of exponential sums, the usual tool to deal with these problems.

Let f(X) Î \mathbb Z[X]{f(X) \in \mathbb {Z}[X]} be an irreducible polynomial of degree D ≥ 2 and let N be a sufficiently large positive integer. We estimate the number of positive integers n ≤ N such that the product F(n) = Õk = 1n f(k)F(n) = \prod\limits_{k =1}^n f(k) is a perfect square. We also consider more general questions and give a low...

For any irreducible quadratic polynomial f(x) in Z[x] we obtain the estimate log l.c.m.(f(1),...,f(n))= n log n + Bn + o(n) where B is a constant depending on f. Comment: 26 pages

We study the sumset A+k·A for the first non trivial case, k=3, where k·A={k·a,a∈A}. We prove that |A+3·A|≥4|A|-4 and that the equality holds only for A={0,1,3}, A={0,1,4}, A=3·{0,⋯,n}∪(3·{0,⋯,n}+1) and all the affine transforms of these sets.

We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length tR1/3 on a circle of radius R, for any given t > 0. In particular we prove that any arc of length (40 + 40 3 √ 10)1/3R1/3 on a circle of radiusR, with R > √ 65, contains at most three lattice points,...

Erd\H os and R\'{e}nyi claimed and Vu proved that for all $h \ge 2$ and for all $\epsilon > 0$, there exists $g = g_h(\epsilon)$ 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 $|A \cap [1,x]| \gg x^{1/h - \epsilon}$. We give two new proofs...

For k prime and A a finite set of integers with |A| ≥ $3(k-1)^2$ (k-1)! we prove that |A+k·A| ≥ (k+1)|A| − ⌈k(k+2)/4⌉ where k·A = {ka:a∈A}. We also describe the sets for which equality holds. Postprint (published version)

Motivated by a question of S\'ark\"ozy, we study the gaps in the product sequence $\B=\A ... \A=\{b_n=a_ia_j, a_i,a_j\in \A\}$ when $\A$ has upper Banach density $\alpha>0$. We prove that there are infinitely many gaps $b_{n+1}-b_n\ll \alpha^{-3}$ and that for $t\ge2$ there are infinitely many $t$-gaps $b_{n+t}-b_{n}\ll t^2\alpha^{-4}$. Furthermore...

Estimating the discrepancy of the set of all arithmetic progressions in the first N natural numbers was one of the famous open problems in combinatorial discrepancy theory for a long time, successfully solved by K. Roth (lower bound) and Beck (upper bound). They proved that D(N)=minχmaxA|∑x∈Aχ(x)|=Θ(N1/4), where the minimum is taken over all colori...

Motivated by a question of S\'ark\"ozy, we study the gaps in the product sequence $\B=\A ... \A=\{b_n=a_ia_j, a_i,a_j\in \A\}$ when $\A$ has upper Banach density $\alpha>0$. We prove that there are infinitely many gaps $b_{n+1}-b_n\ll \alpha^{-3}$ and that for $t\ge2$ there are infinitely many $t$-gaps $b_{n+t}-b_{n}\ll t^2\alpha^{-4}$. Furthermore...

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. Comment: 21 pages, no figures

Let N(a,m) be the least integer n (if it exists) such that.(n) = a (mod m). Friedlander and Shparlinski proved that for any e > 0 there exists A = A(e) > 0 such that for any positive integer m which has no prime divisors p < (logm)A and any integer a with gcd(a,m) = 1, we have the bound N(a,m) ≥ m3+e. In the present paper we improve this bound to N...

When A and B are subsets of the integers in [1, X] and [1, Y], respectively, with |A| ≥ α X and |B| ≥ β Y, we show that the number of rational numbers expressible as a/b with (a, b) in A × B is ≫ (α β)1 + ϵXY for any ϵ > 0, where the implied constant depends on ϵ alone. We then construct examples that show that this bound cannot, in general, be imp...

We prove that, if 2 ≤ k1 ≤ k2, then there is no infinite sequence of positive integers such that the representation function r(n) = (a, a′): n = k1a + k2a′, a, a′ ∈ is constant for n large enough. This result completes the previous work of Dirac and Moser for the special case k1 = 1 and answers a question posed by Sárkozy and Sós.

We show that for a fixed integer base g≥2 the palindromes to base g which are k-powers form a very thin set in the set of all base g palindromes.

A set of integers A is called a B-2[g] set if every integer m has at most g representations of the form m = a + a', with a <= a' and a, a' is an element of A. We obtain a new lower bound for F(g, n), the largest cardinality of a B-2[g] set in {1, ..., n}. More precisely, we prove that lim inf(n ->infinity)F(g, n)/root gn >= 2/root pi - epsilon(g) w...

We prove that the product is a square only for .

We prove that the product ∏k=1n(k2+1) is a square only for n=3n=3.

In this note, we show that the set of n such that the arithmetic mean of the first n primes is an integer is of asymptotic density zero. We use the same method to show that the set of n such that the sum of the first n primes is a square is also of asymptotic density zero. We also prove that both the arithmetic mean of the first n primes as well as...

Let A be a set of integers and let h \geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to the set A, and a_1\leq ... \leq a_h. The function r_{A,h} from the integers Z to the nonnegative integers N_0 U {\infty} is called the representation function of order h f...

We establish a general and optimal lower bound for the complete sum of the probabilities of $k$-intersections of $n$ events. We then describe various applications to additive and multiplicative number theory, graph theory, coding theory, study of lattice points on circles, and divisors of polynomials.

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A set \( \mathcal{A} \) of positive integers is a perfect difference set if every nonzero integer has a unique representation as the difference of two elements of \( \mathcal{A} \). We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set \( \mathcal{...

Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures in harmonic analysis, analytic number theory, arithmetic geometry, discrete geometry and additive combinatorics (some old and some new) which each, if...

We give a non-trivial upper bound for Fh(g,N), the size of a Bh[g] subset of {1,…,N}, when g>1. In particular, we prove F2(g,N)⩽1.864(gN)1/2+1, and Fh(g,N)⩽1(1+cosh(π/h))1/h(hh!gN)1/h, h>2. On the other hand, we exhibit B2[g] subsets of {1,…,N} with g+[g/2]g+2[g/2]N1/2+o(N1/2), elements.

We prove that the lattice points on the circles x2 + y2 = n are well distributed for most circles containing lattice points.

We exhibit, for any integerg≥2, an infinite sequenceA ∈B 2[g] such that limsupx ® ¥ A( x )/Öx = ( Ö{9/8} )Ö{g - 1}\lim \sup _{x \to \infty } A\left( x \right)/\sqrt x = \left( {\sqrt {9/8} } \right)\sqrt {g - 1} . Furthermore, we obtain better estimates for small values ofg. For instance, we exhibit an infinite sequenceA ∈B 2[2] such that limsupx...

Let E be a semistable, rank two vector bundle of degree d on a Riemann surface C of genus g≥1, i.e. such that the minimal degree s of a tensor product of E with a line bundle having a nonzero section is nonnegative. We give an analogue of Clifford’s lemma by showing that E has at most (d-s)/2+δ independent sections, where δ is 2 or 1 according to w...

Let Fh(N) be the maximum number of elements that can be selected from the set {1, …, N} such that all the sums a1+…+ah, a1⩽…⩽ah are different. We introduce new combinatorial and analytic ideas to prove new upper bounds for Fh(N). In particular we proveF3(N)⩽41+16/(π+2)4N1/3+o(N1/3),F4(N)⩽81+16/(π+2)4N1/4+o(N1/4).Besides, our techniques have an inde...

New upper and lower bounds are given for Fh(g, N), the maximum size of a Bh[g] sequence contained in [1, N]. It is proved that and that

We introduce a new counting method to deal with B2[2] sequences, getting a new upper bound for the size of these sequences, F(N, 2)⩽6N+1.

Abstract

We include several results providing bounds for an interval on the hyperbola xy = N containing k lattice points.

In this paper we find an upper bound for the number of lattice points on an arc of small length on the conicsx2−dy2=N. Our method involves factorization into ideals in quadratic fields.

Let k ≥ 2 be an integer. For fixed N, we consider a set AN of non-negative integers such that for all integer n ≤ N, n can be written as n = a + bk, a ∈ AN, b a positive integer. We are interested in a lower bound for the number of elements of AN. Improving a result of R. Balasubramanian (J. Number Theory29, 1988, 10-12), we prove the following the...

In this paper we study the distribution of lattice points on arcs of circles centered at the origin. We show that on such a circle of radius R, an arc whose length is smaller than $\sqrt2 R^{1/2 - 1(4\lbrack m/2 \rbrack + 2)}$ contains, at most, m lattice points. We use the same method to obtain sharp L4-estimates for uncompleted, Gaussian sums.

Motivated by a question of S arkozy, we study the gaps in the product sequence B =AA =fbn = aiaj; ai;aj2Ag whenA has upper Banach density > 0. We prove that there are innitely many gaps bn+1 bn 3 and that for t 2 there are innitely many t-gaps bn+t bn t2 4. Furthermore we prove that these estimates are best possible. We also discuss a related quest...

Uno de los problemas favoritos de Erdős, y que mejor ha descrito su gusto por la «aritmética combinatoria», ha sido el de los conjuntos de Sidon. La referencia a «El Libro» donde se encuentran las de-mostraciones más hermosas es constante en la obra de Erdős. Corría el año 1932 cuando Simon Sidon, el ana-lista húngaro, le preguntó a Erdős sobre con...

Tesis doctoral inédita. Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 14-7-1995 Bibliografía: h. 67-69