An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $μ^\ast$ defined on the whole power set of $\mathbf H$ such that $μ^\ast(X) \le μ^\ast({\bf H}) = 1$ and $μ^\ast(k \cdot X + h) = \frac{1}{k}\, μ^\ast(X)$ for all $X \subseteq \bf H$, $k \in {\bf N}^+$, and $h \in \bf N$, where $k \cdot X := \{kx: x \in X\}$; and an upper density on $\bf H$ is an upper quasi-density on $\bf H$ that is non-decreasing with respect to inclusion. We say that a set $X \subseteq \bf H$ is small if $μ^\ast(X) = 0$ for every upper quasi-density $μ^\ast$ on $\bf H$.
Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper Pólya densities, along with the uncountable family of upper $α$-densities, where $α$ is a real parameter $\ge -1$ (most notably, $α= -1$ corresponds to the upper logarithmic density, and $α= 0$ to the upper asymptotic density).
It turns out that a subset of $\bf H$ is small if and only if it belongs to the zero set of the upper Buck density on $\bf Z$. This allows us to show that many interesting sets are small, including the integers with less than a fixed number of prime factors, counted with multiplicity; the numbers represented by a binary quadratic form with integer coefficients whose discriminant is not a perfect square; and the image of $\bf Z$ through a non-linear integral polynomial in one variable.
We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.
The study of the additive volume of sets can be reduced to the case of one-dimensional sets. The exact values of the volume of extremal sets are given as a conjecture.
Consider the sets of integers $A$ that avoid any arrangement of $g$ congruent $h$-subsets. Our findings refine and improve upon some results by Erdős and Harzheim about these sets.
Several measures for the density of sets of integers have been proposed, such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and investigate some of its properties.
We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
We consider two number-theoretic problems arising from Fuglede's spectral set conjecture: characterizing finite sets that tile integers, and finding polynomials with (0,1) coefficients whose roots have a certain multiplicative structure. We verify several special cases of the relevant conjectures; in particular, we find necessary and sufficient conditions for a set A to tile the integers if A is a direct sum of cyclic subsets.
We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.
We consider the problem of characterizing finite sets which tile the integers by translations. Coven and Meyerowitz (J. Algebra 1999) found necessary and sufficient conditions for a finite set A to tile the integers under the assumption that |A| has at most 2 distinct prime factors. The purpose of this article is to settle, for the first time, certain three-prime cases.
We discuss the problem of finding distinct integer sets $\{x_1,x_2,...,x_n\}$ where each sum $x_i+x_j, i \ne j$ is a square, and $n \le 7$. We confirm minimal results of Lagrange and Nicolas for $n=5$ and for the related problem with triples. We provide new solution sets for $n=6$ to add to the single known set. This provides new information for problem D15 in Guy's {\it Unsolved Problems in Number Theory}
Let A be a set of integers dense in a finite interval. We establish upper and lower bounds for the longest regularly-spaced and convex subsets of A and of A-A.
We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets which do not contain aritmetic progressions of length 3.
We show that if there exists an integer subject to some congruence conditions that cannot be written as the sum of the norm of an ideal in $\mathbb{Z}[\exp(2πi/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then there are infinitely many such integers. Also, if there exists an integer that cannot be written as the sum of an integer which is the norm of an ideal in in $\mathbb{Z}[\exp(2πi/p)]$ and at most $p-2$ powers of $p$, where $p\geq 3$ is a prime, then there are infinitely many such integers. Finally it is shown that there are infinitely many integers not the sum of the norm of an ideal…
We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erdös and Szemerédi on sum and product sets.
We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets are bases of finite order for the positive integers. We show that generically, $A$ is a basis of order $2$ when $\operatorname{deg} p \geq 3$, but not when $\operatorname{deg} p = 2$, although then $A + A$ still has asymptotic density $1$.
For nonempty sets $A,B$ of nonnegative integers and an integer $n$, let $r_{A,B}(n)$ be the number of representations of $n$ as $a+b$ and $d_{A,B}(n)$ be the number of representations of $n$ as $a-b$, where $a\in A, b\in B$. In this paper, we determine the sets $A,B$ such that $r_{A,B}(n)=1$ for every nonnegative integer $n$. We also consider the \emph{difference} structure and prove that: there exist sets $A$ and $B$ of nonnegative integers such that $r_{A,B}(n)\ge 1$ for all large $n$, $A(x)B(x)=(1+o(1))x$ and for any given nonnegative integer $c$, we have $d_{A,B}(n)=c$ for infinitely many …
Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y and g(x,y)=u_2x+v_2y with integral coefficients, there exist arbitrarily large finite sets of integers A and B such that |f(A)| > |g(A)| and |f(B)| < |g(B)|.
In this paper, we shall prove that any subset of $\overline{\mathbb Q}\cap B(0,1)$, which is closed under complex conjugation and which contains the element $0$, is the exceptional set of uncountably many transcendental functions, analytic in the unit ball, with integer coefficients. This solves a strong version of an old question proposed by K. Mahler (1976).
Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a sum of $h$ elements of $\mathcal{A}$. The structure of the set $(h\mathcal{A})^{(t)}$ is completely determined for all $h \geq h_t$.
Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let the representation function $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. In this paper, we determine the structure of $C, D\subseteq \mathbb{N}$ with $C\cup D=[0, m]$ and $|C\cap D|=2$ such that $R_{C}(n)=R_{D}(n)$ for any nonnegative integer $n$.
If, for a subset S of Z^k, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and, (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case) then (a) implies (b) (obviously), (b) implies (c), and neither converse holds. Condition (b) is equivalent to the set S being the set of integer values taken by some k-tuple of polynomials with rational coefficients as the variables range through the integers. We also show…
We study the typical behavior of the least common multiple of the elements of a random subset $A\subset \{1,\dots, n\}$. For example we prove that $\text{lcm}\{a:\ a\in A\}=2^{n(1+o(1))}$ for almost all subsets $A\subset\{1,\dots,n\}$.
In [10] the third author of this paper presented two conjectures on the additive decomposability of the sequence of ''smooth'' (or ''friable'') numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of $S$-unit equations).
In this paper we show that any Bohr-zero non-periodic set $B$ of traceless integer valued matrices, denoted by $Λ$, intersects non-trivially the conjugacy class of any matrix from $Λ$. As a corollary, we obtain that the family of characteristic polynomials of $B$ contains all characteristic polynomials of matrices from $Λ$. The main ingredient used in this paper is an equidistribution result for an $SL_d(\mathbf{Z})$ random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work.
In part I of this paper we studied additive decomposability of the set $\F_y$ of th $y$-smooth numbers and the multiplicative decomposability of the shifted set $\g_y=\F_y+\{1\}$. In this paper, focusing on the case of 'large' functions $y$, we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of $k$-term sumsets, for arbitrary $k$.
Let $a_0=b_0=0$ and $0<a_1\leq b_1<a_2\leq b_2<\ldots\leq b_{n}$ be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in $\bigcup_{j=1}^{n}[a_j,b_j]$. The following formula holds: $$\lim_{x\to\infty}{\frac{Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)}{x}}=\prod\limits_{p}\sum\limits_{i=0}^{n}\left(\frac{1}{p^{a_{i}}}-\frac{1}{p^{b_{i}+1}}\right).$$ In this paper, we prove this result and then generalize it.
When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq αX$ and $|B| \geq βX$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is $\gg (αβ)^{1+ε}XY$ 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 improved to $\gg αβXY$. We also resolve the natural generalisation of our problem to arbitrary subsets $C$ of the integer points in $[1,X] \times [1,Y]$. Finally, we apply our results to answer a question of Sárközy concernin…
A set of integers $S \subset \mathbb{N}$ is an $α$-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on $α$, more specifically if $| (x+w) - (y+z) | \geq \max \{ x^α,y^α,z^α,w^α\}$ for every $x,y,z,w \in S$ satisfying $\max \{x,w\} \neq \max \{y,z\}$. We obtain a new lower bound for the growth of $α$-strong infinite Sidon sets when $0 \leq α< 1$. We also further extend that notion in a natural way by obtaining the first non-trivial bound for $α$-strong infinite $B_h$ sets. In both cases, we study the implications of these bounds for the density …
Fix $α\in (0,1/3)$. We show that, from a topological point of view, almost all sets $A\subseteq \mathbb{N}$ have the property that, if $A^\prime=A$ for all but $o(n^α)$ elements, then $A^\prime$ is not a nontrivial sumset $B+C$. In particular, almost all $A$ are totally irreducible. In addition, we prove that the measure analogue holds with $α=1$.
Let $h,k \ge 2$ be integers. A set $A$ of positive integers is called asymptotic basis of order $k$ if every large enough positive integer can be written as the sum of $k$ terms from $A$. A set of positive integers $A$ is said to be a $B_{h}[g]$-set if every positive integer can be written as the sum of $h$ terms from $A$ at most $g$ different ways. In this paper we prove the existence of $B_{h}[1]$ sets which are asymptotic bases of order $2h$ by using probabilistic methods.
A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this size we show that, under mild assumptions on the size of the modulus $m$ and on the difference $N^{1/2}-\Abs{\cal A}$ (these quantities should not be too large) the elements of ${\cal A}$ are uniformly distributed in the residue classes mod $m$. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lin…
A subset $C$ of an abelian group $G$ is a minimal additive complement to $W \subseteq G$ if $C + W = G$ and if $C' + W \neq G$ for any proper subset $C' \subset C$. In this paper, we study which sets of integers arise as minimal additive complements. We confirm a conjecture of Kwon, showing that bounded-below sets with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences. We bound the upper and lower Banach density of syndetic sets that aris…
Let f(x_1,x_2,...,x_m) = u_1x_1+u_2 x_2+... + u_mx_m be a linear form with positive integer coefficients, and let N_f(k) = min{|f(A)| : A \subseteq Z and |A|=k}. A minimizing k-set for f is a set A such that |A|=k and |f(A)| = N_f(k). A finite sequence (u_1, u_2,...,u_m) of positive integers is called complete if {\sum_{j\in J} u_j : J \subseteq {1,2,..,m}} = {0,1,2,..., U}, where $U = \sum_{j=1}^m u_j.$ It is proved that if f is an m-ary linear form whose coefficient sequence (u_1,...,u_m) is complete, then N_f(k) = Uk-U+1 and the minimizing k-sets are precisely the arithmetic progressions of…
Let $φ(x_1,\ldots,x_h,y) = u_1x_1 + \cdots + u_hx_h+vy$ be a linear form with nonzero integer coefficients $u_1,\ldots, u_h, v.$ Let $\mathcal{A} = (A_1,\ldots, A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an infinite set of integers. Define the representation function associated to the form $φ$ and the sets \mca\ and $B$ as follows: $$ R^{(φ)}_{\mathcal{A},B}(n) = \text{card}\left( \left\{ (a_1,\ldots, a_h,b) \in A_1 \times \cdots \times A_h \times B: φ(a_1, \ldots , a_h,b ) = n \right\} \right).$$ If this representation function is constant, then the set $B$ is periodic an…
Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leqα\leqβ\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that $\mathrm{d}(A)=α$ and $\mathrm{d}(A+A)=β$. More generally we study the set of $k$-tuples $(\mathrm{d}(iA))_{1\leq i\leq k}$ for $A\subset \mathbb{N}$. This leads us to introduce subsets defined by diophantine constraints inside a random set of integers known as the set of ``pseudo $s$th powers''. We consider similar problems for subsets of the circle $\mathbb{R}/\mathbb{Z}$, …
If f is a polynomial with integer coefficients and q is an integer, we may regard f as a map from Z/qZ to Z/qZ. We show that the distribution of the (normalized) spacings between consecutive elements in the image of these maps becomes Poissonian as q tends to infinity along any sequence of square free integers such that the mean spacing modulo q tends to infinity.
A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B_2[g]-set is a set A of integers such that no integer has more than g essentially distinct representations as the sum of two elements of A. It is proved that almost all small sumsets of {1,2,...,n} are B_2[g]-sets, in the sense that if B_2[g](k,n) denotes the number of B_2[g]-sets of cardinality k contained in the interval {1,2,...,n}, then lim_{n\to\infty} B_2[g](k,n)/\binom{n}{k} = 1 if k = o(n^{g/(2g+2)}).
Inspired by the Erdös-Turan conjecture we consider subsets of the natural numbers that contains infinitely many aritmetic progressions (APs) of any given length - such sets will be called AP-sets and we know due to the Green-Tao Theorem and Sźmeredis Theorem that the primes and all subsets of positive upper density are AP-sets. We prove that (1, 1,..., 1) is a solution to the equation Mx = 0 where M is an integer matrix whose null space has dimension at least 2, if and only if the equation has infinitely many solutions such that the coordinates of each solution are elements in the same AP. Thi…
Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum is extended over the $S$-divisors of $n$. We determine the sets $S$ such that the $S$-convolution is associative and preserves the multiplicativity of functions, respectively, and discuss other basic properties of it. We give asymptotic formulae with error terms for the functions $σ_S(n)$ and $τ_S(n)$, representing the sum and the number of $S$-divisors of $…
A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{1,\ldots,n\}$ for which the integer $2n+1$ cannot be represented as a sum of their elements. We prove a bound of $O(2^{n/3})$ on the number of these sets, which matches, up to a multiplicative constant, the lower bound obtained by considering all subsets of $B_n = \{ \lceil \frac{2}{3}(n+1) \rceil, \ldots, n \}$. A main ingredient in the proof is a stability theorem saying that if a subset of $[n]$ of size close to $|B_n|$ contains only a few subsets that co…
