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.
Abstract upper densities are monotone and subadditive functions from the power set of positive integers to the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density. We answer a question posed by G. Grekos in 2013, and prove the existence of translation invariant abstract upper densities onto the unit interval, whose null sets are precisely the family of finite sets, or the family of sequences whose series of reciprocals converge. We also show that no such density can be …
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.
We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if $K$ is a number field and $S$ is any set of prime ideals with natural density $δ(S)$ within the primes, then \[ -\lim_{X \to \infty}\sum_{\substack{2 \le \operatorname{N}(\mathfrak{a})\le X\\ \mathfrak{a} \in D(K,S)}}\frac{μ(\mathfrak{a})}{\operatorname{N}(\mathfrak{a})} = δ(S), \] where $μ(\mathfrak{a})$ is the generalized Möbius function and $D(K,S)$ is the set of integral ideals $ \mathfrak{a} \subseteq \mathcal{O}_K$ with unique prime divisor of min…
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 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}
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…
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 …
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 $\mathbb{N}$ 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 $n$.
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).
We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{\mathrm{min}}(n)$ of integers $n\geq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $\mathbb{Q}$. Then we prove that $$-\lim_{X\rightarrow\infty}\sum_{\substack{2\leq n\leq X\\[1pt]\left[\frac{K/\mathbb{Q}}{p_{\mathrm{min}}(n)}\right]=C}}\frac{μ(n)}{n}=\frac{\#C}{\#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{\mathr…
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.
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 $k$.
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…
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 $\gcd(r,t)=1$, then a theorem of Alladi offers the Möbius sum identity $$-\sum_{\substack{ n \geq 2 \\ p_{\rm{min}}(n) \equiv r \pmod{t}}} μ(n)n^{-1}= \frac{1}{φ(t)}. $$ Here $p_{\rm{min}}(n)$ is the smallest prime divisor of $n$. The right-hand side represents the proportion of primes in a fixed arithmetic progression modulo $t$. Locus generalized this to Chebotarev densities for Galois extensions. Answering a question of Alladi, we obtain analogs of these results to arithmetic densities of subsets of positive integers using $q$-series and integer partitions. For suitable subsets $§$ of th…
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.
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 $…
Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) = \text{max}_{A \in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erdős conjectured that $f(n, k) = |E(n,k)|$ for every $n \ge p_{k}$. Recently Chen and Z…
For a set of nonnegative integers $A$, denote by $R_{A}(n)$ the number of unordered representations of the integer $n$ as the sum of two different terms from $A$. In this paper we partially describe the structure of the sets, which have coinciding representation functions.
Let $B$ be a finite set of natural numbers or complex numbers. Product set corresponding to $B$ is defined by $B.B:=\{ab:a,b\in B\}$. In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when $B$ is a set of natural numbers and a bound which is accurate up to a positive constant when $B$ is a set of complex numbers.
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.
An MSTD set is a finite set of integers with more sums than differences. It is proved that, for infinitely many positive integers $k$, there are infinitely many affinely inequivalent MSTD sets of cardinality $k$. There are several related open problems.
The Erdos-Davenport theorem on the multiples claims that for any set of natural numbers the set consisting of their multiples possesses the logarithmic density. An analogous statement is proved for the sets of rational multiples.
Expansions in non-integer bases have been investigated abundantly since their introduction by Rényi. It was discovered by Erdős et al. that the sets of numbers with a unique expansion have a much more complex structure than in the integer base case. The present paper is devoted to the continuity properties of these maps with respect to the Hausdorff metric.
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.
Given two sets of positive integers $A$ and $B$, let $AB := \{ab : a \in A,\, b \in B\}$ be their product set and put $A^k := A \cdots A$ ($k$ times $A$) for any positive integer $k$. Moreover, for every positive integer $n$ and every $α\in [0,1]$, let $\mathcal{B}(n, α)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $α$. We prove that if $A_1, \dots, A_s$ are random sets in $\mathcal{B}(n_1, α_1), \dots, \mathcal{B}(n_s, α_s)$, respectively, $k_1, \dots, k_s$ are f…
Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of $a_1+h_1, \ldots, a_n+h_n$ is very different from that of $a_1, \ldots, a_n$. We also consider a similar problem for the least common multiple.
Direction sets, recently introduced by Leonetti and Sanna, are generalization of ratio sets of subsets of positive integers. In this article, we generalize the notion of direction sets and define {\it $k$-generalized direction sets} and {\it distinct $k$-generalized direction sets} for subsets of positive integers. We prove a necessary condition for a subset of $\mathcal{S}^{k - 1} := \{\underline{x} \in [0,1]^{k} : ||\underline{x}|| = 1\}$ to be realized as the set of accumulation points of a distinct $k$-generalized direction set. We provide sufficient conditions for some particular subsets …
The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in $\C^d.$ We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the Hilbert-Fekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single variable polynomials in the complex plane, our estimate coincides with the Hilbert-Fekete result.
A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct a strictly decreasing sequence $(a_i)_{i=1}^{\infty}$ of positive real numbers with $a_1 = 1$ such that the set \[ G^{(k)} = \bigcup_{i=1}^{\infty} \left(a_{2i} , a_{2i-1} \right] \] contains no geometric progression of length $k$ and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of $(0,1]$ that contains no geometric progression of length $k$ and in…
We construct an easily described family of partitions of the positive integers into $n$ disjoint sets with essentially the same structure for every $n \geq 2$. In a special case, it interpolates between the Beatty $\frac{1}ϕ + \frac{1}{ϕ^2} = 1$ partitioning ($n=2$) and the 2-adic partitioning in the limit as $n \rightarrow \infty$. We then analyze how membership of elements in the sets of one partition relates to membership in the sets of another. We investigate in detail the interactions of two Beatty partitions with one another and the interactions of the $ϕ$ Beatty partition mentioned abov…
We consider in this paper the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-theoretical extensions of some classical sumset theorems in compact abelian groups due to Kneser. Our proofs are based on a correspondence principle for action sets which was developed previously by the first two authors.
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 prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic progression of length |2S|-|S|+1. This is the first result of this nature which places no unnecessary restrictions on the size of S.
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 $1$.
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)|.
