Combinatorial and Additive Number Theory IV

CANT, New York, USA, 2019 and 2020

2021
Book

Editor: Melvyn B. Nathanson
Book Series: Springer Proceedings in Mathematics & Statistics
Publisher: Springer International Publishing
Part of: Springer Professional "Business + Economics & Engineering + Technology"; Springer Professional "Engineering + Technology"; Springer Professional "Business + Economics"

About this book

This is the fourth in a series of proceedings of the Combinatorial and Additive Number Theory (CANT) conferences, based on talks from the 2019 and 2020 workshops at the City University of New York. The latter was held online due to the COVID-19 pandemic, and featured speakers from North and South America, Europe, and Asia. The 2020 Zoom conference was the largest CANT conference in terms of the number of both lectures and participants.
These proceedings contain 25 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003 at the CUNY Graduate Center, the workshop surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, zero-sum sequences, minimal complements, analytic and prime number theory, Hausdorff dimension, combinatorial and discrete geometry, and Ramsey theory. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Frontmatter

Extremal Sequences for Some Weighted Zero-Sum Constants for Cyclic Groups

Abstract

A particular weighted generalization of some classical zero-sum constants became popular and some applications of this weighted generalization have also been found. After some introductory remarks, we here take up some questions regarding inverse problems related to the values of a weighted zero-sum constant for some particular weights for a finite cyclic group.

S. D. Adhikari, Md Ibrahim Molla, Shameek Paul

On a Zero-Sum Problem Arising From Factorization Theory

Abstract

We study a zero-sum problem dealing with minimal zero-sum sequences of maximal length over finite abelian groups. A positive answer to this problem yields a structural description of sets of lengths with maximal elasticity in transfer Krull monoids over finite abelian groups.

Aqsa Bashir, Alfred Geroldinger, Qinghai Zhong

Conditional Bounds on Siegel Zeros

Abstract

We present an overview of bounds on zeros of L-functions and obtain some improvements under weak conjectures related to the Goldbach problem.

Gautami Bhowmik, Karin Halupczok

Infinite Co-minimal Pairs in the Integers and Integral Lattices

Abstract

Given two nonempty subsets A, B of a group G, they are said to form a co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any $\emptyset \ne A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \ne B' \subsetneq B$. The existence of co-minimal pairs is a stronger criterion than the existence of minimal complements. In this work, we show several new results about them. The existence and the construction of co-minimal pairs in the integers, with both the subsets A and B (A is not a translate of B) of infinite cardinality was unknown. We show that such pairs exist and give the first explicit construction of these pairs. The constructions also satisfy a number of algebraic properties. Further, we prove that for any $d\ge 1$, the group $\mathbb {Z}^{2d}$ admits infinitely many automorphisms such that for each such automorphism $\sigma $, there exists a subset A of $\mathbb {Z}^{2d}$ such that A and $\sigma (A)$ form a co-minimal pair.

Arindam Biswas, Jyoti Prakash Saha

Rigidity, Graphs and Hausdorff Dimension

Abstract

A set of $k+1$ points in Euclidean space is called a $(k+1)$-point configuration. Two configurations are congruent if they are equal up to an affine isometry. Given a compact subset E of $\mathbb R^d$, $d\ge 2$ of Hausdorff dimension greater than $d-\frac{1}{k+1}$ we prove that the Lebesgue measure of noncongruent $(k+1)$-point configurations in E is positive, for $k>d$, complementing the results of [11] for $k\le d$.

Nikolaos Chatzikonstantinou, Alex Iosevich, Sevak Mkrtchyan, Jonathan Pakianathan

On Generalized Harmonic Numbers

Abstract

For three positive integers a, b and n, let $H_{a,b}(n)$ be the sum of the reciprocals of the first n terms of arithmetic progression $\{ ak+b : k=0,1, \ldots \} $ and let $v_{a,b} (n)$ be the denominator of $H_{a,b}(n).$ In this paper, we prove that for two coprime positive integers a and b, (i) if p is a prime with $p\not \mid a$, then the set of positive integers n with $p\mid v_{a,b} (n)$ has asymptotic density one; (ii) the set of positive integers n with $v_{a,b} (n)=v_{a,b} (n+1)$ has asymptotic density one.

Yong-Gao Chen, Bing-Ling Wu

Partitions for Semi-magic Squares of Size Three

Abstract

In the theory of Clebsch-Gordan coefficients, one may recognize the domain space as the set of weakly semi-magic squares of size three. Two partitions on this set are considered: a triangle-hexagon model based on top lines, and one based on the orbits under a finite group action. In addition to giving another proof of McMahon’s formula, we give a generating function that counts the so-called trivial zeros of Clebsch-Gordan coefficients and its associated quasi-polynomial.

Robert W. Donley

A Sum of Negative Degrees of the Gaps Values in Two-Generated Numerical Semigroups and Identities for the Hurwitz Zeta Function

Abstract

We derive an explicit expression for an inverse power series over the gaps values of numerical semigroups generated by two integers. It implies the multiplication theorem for the Hurwitz zeta function $\zeta (n,q)$.

Leonid G. Fel, Takao Komatsu, Ade Irma Suriajaya

Widely Digitally Stable Numbers

Abstract

We show that there are infinitely many composite numbers N, relatively prime to 10, that remain composite if you insert any digit anywhere in its base 10 representation, including between two of the infinitely many leading zeros of N and to the right of the units digit of N.

Michael Filaseta, Jacob Juillerat, Jeremiah Southwick

Non-injectivity of Nonzero Discriminant Polynomials and Applications to Packing Polynomials

Abstract

We show that an integer-valued quadratic polynomial on $\mathbb {R}^2$ can not be injective on the integer lattice points of any affine convex cone if its discriminant is nonzero. A consequence is the non-existence of quadratic packing polynomials on irrational sectors of $\mathbb {R}^2$.

Kåre S. Gjaldbæk

Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

For a sequence S of terms from an abelian group G of length |S|, let $\Sigma _n(S)$ denote the set of all elements that can be represented as the sum of terms in some n-term subsequence of S. When the subsum set is very small, $|\Sigma _n(S)|\le |S|-n+1$, it is known that the terms of S can be partitioned into n nonempty sets $A_1,\ldots ,A_n\subseteq G$ such that $\Sigma _n(S)=A_1+\ldots +A_n$. Moreover, if the upper bound is strict, then $|A_i\setminus Z|\le 1$ for all i, where $Z=\bigcap _{i=1}^{n}(A_i+H)$ and $H=\{g\in G:\; g+\Sigma _n(S)=\Sigma _n(S)\}$ is the stabilizer of $\Sigma _n(S)$. This allows structural results for sumsets to be used to study the subsum set $\Sigma _n(S)$ and is one of the two main ways to derive the natural subsum analog of Kneser’s Theorem for sumsets. In this paper, we show that such a partitioning can be achieved with sets $A_i$ of as near equal a size as possible, so $\lfloor \frac{|S|}{n}\rfloor \le |A_i|\le \lceil \frac{|S|}{n}\rceil $ for all i, apart from one highly structured counterexample when $|\Sigma _n(S)|= |S|-n+1$ with $n=2$. The added information of knowing the sets $A_i$ are of near equal size can be of use when applying the aforementioned partitioning result, or when applying sumset results to study $\Sigma _n(S)$ (e.g., [20]). We also give an extension increasing the flexibility of the aforementioned partitioning result and prove some stronger results when $n\ge \frac{1}{2}|S|$ is very large.

David J. Grynkiewicz

Bounds on Point Configurations Determined by Distances and Dot Products

Abstract

We study a family of variants of Erdő’s unit distance problem, concerning distances and dot products between pairs of points chosen from a large finite point set. Specifically, given a large finite set of n points E, we look for bounds on how many subsets of k points satisfy a set of relationships between point pairs based on distances or dot products. We survey some of the recent work in the area and present several new, more general families of bounds.

Slade Gunter, Eyvi Palsson, Ben Rhodes, Steven Senger

Distribution of Missing Differences in Diffsets

Abstract

Lazarev, Miller and O’Bryant [11] investigated the distribution of $|S+S|$ for S chosen uniformly at random from $\{0, 1, \dots , n-1\}$, and proved the existence of a divot at missing 7 sums (the probability of missing exactly 7 sums is less than missing 6 or missing 8 sums). We study related questions for $|S-S|$, and show some divots from one end of the probability distribution, $P(|S-S|=k)$, as well as a peak at $k=4$ from the other end, $P(2n-1-|S-S|=k)$. A corollary of our results is an asymptotic bound for the number of complete rulers of length n.

Scott Harvey-Arnold, Steven J. Miller, Fei Peng

Recent Progress in Hilbert Cubes Theory

Abstract

In 1978 Nathanson obtained several results on sumsets contained in infinite sets of integers. Later the author investigated how big a Hilbert cube avoiding a given infinite sequence of integers can be. In the present paper we concentrate on Densities and Hilbert cubes, Hilbert cubes which avoid given sets, and $\Delta $-degenerate Hilbert cubes. The aim is to collect some results in the past and some related recent problems.

Norbert Hegyvári

Intrinsic Characterization of Representation Functions and Generalizations

Abstract

Given a set A of natural numbers, i.e., nonnegative integers, we establish an intrinsic characterization of the representation function of A, which to every natural number n associates the number $r_A(n)$ of ordered pairs (a, b) of elements $a, b \in A$ such that $a+b=n$, thus answering an open problem stated many years ago by Mel Nathanson. We also establish similar characterizations of the characteristic function $\chi _{A}(n)$, which is equal to 1 or 0 according as the natural number n lies or does not lie in A, and the counting function A(n), which gives the number of elements a of A satisfying $a\le n$. We then generalize to representation functions as sums of more than two elements of A.

Charles Helou

Combinatorics of Multicompositions

Abstract

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing zero as some parts were introduced by Ouvry and Polychronakos in 2019. We give a bijection between these two varieties of compositions and determine various combinatorial properties of these multicompositions. In particular, we determine the count of multicompositions by number of all parts, number of positive parts, and number of zeros. Then, working from three types of compositions with restricted parts that are counted by the Fibonacci sequence, we find the sequences counting multicompositions with analogous restrictions. With these tools, we give combinatorial proofs of summation formulas for generalizations of the Jacobsthal and Pell sequences.

Brian Hopkins, Stéphane Ouvry

On the Connection Between the Goldbach Conjecture and the Elliott-Halberstam Conjecture

Abstract

In this paper we prove that the binary Goldbach conjecture for sufficiently large even integers would follow under the assumptions of both the Elliott-Halberstam conjecture and a variant of the Elliott-Halberstam conjecture twisted by the Möbius function, provided that the sum of their level of distributions exceeds 1. This continues the work of Pan [10]. An analogous result for the twin prime conjecture is obtained by Ram Murty and Vatwani [13].

Jing-Jing Huang, Huixi Li

Part-Frequency Matrices, II: Recent Work

Abstract

We illustrate the use of part-frequency matrices as a tool for combinatorial proofs of partition theorems. Several new theorems are rapidly proved, and progress is made toward replacing the proof of a known theorem proved by modular forms with a more combinatorial approach. A few lines of investigation are suggested. All currently-published papers known to the author that employ the idea are cited here.

William J. Keith

A Conjectural Inequality for Visible Points in Lattice Parallelograms

Abstract

Let $a,n \in \mathbb Z^+$, with $a<n$ and $\gcd (a,n)=1$. Let $P_{a,n}$ denote the lattice parallelogram spanned by (1, 0) and (a, n), that is,

$$P_{a,n} = \left\{ t_1(1,0)+ t_2(a,n) \, : \, 0\le t_1,t_2 \le 1 \right\} , $$

and let

$$V(a,n) = \# \text { of visible lattice points in the interior of } P_{a,n}.$$

In this paper we prove some elementary (and straightforward) results for V(a, n). The most interesting aspects of the paper are in Section 5 where we discuss some numerics and display some graphs of V(a, n)/n. (These graphs resemble an integral sign that has been rotated counter-clockwise by $90^\circ $.) The numerics and graphs suggest the conjecture that for $a\not = 1, n-1$, V(a, n)/n satisfies the inequality

$$ 0.5< V(a,n)/n< 0.75.$$

Gabriel Khan, Mizan R. Khan, Joydip Saha, Peng Zhao

On a Two-Dimensional Exponential Sum

Abstract

In this paper we study the two-dimensional analog of the classical quadratic Weyl sum. We establish a major arc approximation with a strong error term akin to celebrated results of R.C. Vaughan from the early 1980s and the late 2000s. Our main result has applications to the study of discrete maximal operators related to triangular configurations.

Angel V. Kumchev

On Consecutive Perfect Powers with Elementary Methods

Abstract

Catalan’s conjecture states that the equation $x^p-y^q=1$ admits the unique solution $3^2-2^3=1$ in integers $x,y,p,q \ge 2$. The conjecture has been proved by Mihăilescu in 2002 using the theory of cyclotomic fields and Galois modules. Here, relying only on elementary methods, we prove several instances of this result. In particular, we show it in the following cases: p even, q is even, x divides q, y divides $x-1$, y is a power of a prime, and $y\le p^{p/2}$.

Paolo Leonetti

Sidon Sets and Perturbations

Abstract

A subset A of an additive abelian group is an h-Sidon set if every element in the h-fold sumset hA has a unique representation as the sum of h not necessarily distinct elements of A. Let $\mathbf {F}$ be a field of characteristic 0 with a nontrivial absolute value, and let $A = \{a_i :i \in \mathbf {N} \}$ and $B = \{b_i :i \in \mathbf {N} \}$ be subsets of $\mathbf {F}$. Let $\varepsilon = \{ \varepsilon _i:i \in \mathbf {N} \}$, where $\varepsilon _i > 0$ for all $i \in \mathbf {N}$. The set B is an $\varepsilon $-perturbation of A if $|b_i-a_i| < \varepsilon _i$ for all $i \in \mathbf {N}$. It is proved that, for every $\varepsilon = \{ \varepsilon _i:i \in \mathbf {N} \}$ with $\varepsilon _i > 0$, every set $A = \{a_i :i \in \mathbf {N} \}$ has an $\varepsilon $-perturbation B that is an h-Sidon set. This result extends to sets of vectors in $ \mathbf{F}^n$.

Melvyn B. Nathanson

Multiplicative Representations of Integers and Ramsey’s Theorem

Abstract

Let $ \mathcal B= (B_1,\ldots , B_h)$ be an h-tuple of sets of positive integers. Let $g_{ \mathcal B}(n)$ count the number of representations of n in the form $n = b_1\cdots b_h$, where $b_i \in B_i$ for all $i \in \{1,\ldots , h\}$. It is proved that $\liminf _{n\rightarrow \infty } g_{ \mathcal B}(n) \ge 2$ implies $\limsup _{n\rightarrow \infty } g_{ \mathcal B}(n) = \infty $.

Melvyn B. Nathanson

On Distinct Consecutive Differences

Abstract

We show that if $A=\{a_1< a_2< \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb {R}$,

$$ |A+B|\gg |A|^{1/2}|B|. $$

The bound is tight up to the constant.

Imre Ruzsa, George Shakan, József Solymosi, Endre Szemerédi

Limit Points of Nathanson’s Lambda Sequences

Abstract

We consider the set $A_{n}=\displaystyle \cup _{j=0}^{\infty }\{\varepsilon _{j}(n)\cdot n^j:\varepsilon _{j}(n)\in \{0,\pm 1,\pm 2,\ldots ,\pm \lfloor {{n}/{2}}\rfloor \}\} $. Let $\mathscr {S}_{\mathscr {A}}= \bigcup _{a \in \mathscr {A} } A_{a}$ where $\mathscr {A}\subseteq \mathbb {N}$. We denote by $\lambda _{\mathscr {A}}(h)$ the smallest positive integer that can be represented as a sum of h, and no less than h, elements in $\mathscr {S}_{\mathscr {A}}$. Nathanson studied the properties of the $\lambda _\mathscr {A}(h)$-sequence and posed the problem of finding the values of $\lambda _\mathscr {A}(h)$. When $\mathscr {A}=\{2,i\}$, we represent $\lambda _{\mathscr {A}}(h)$ by $\lambda _{2,i}(h)$. Only the values $\lambda _{2,3}(1)=1$, $\lambda _{2,3}(2)=5$, $\lambda _{2,3}(3)=21$ and $\lambda _{2,3}(4)=150$ are known. In this paper, we extend this result. For odd $i>1$ and $h\in \{1,2,3\}$, we find an extensive set of values for $\lambda _{2,i}(h)$. Furthermore, for fixed $h\in \{1,2,3\}$, we find certain values of $\lambda _{2,i}(h)$ that occur infinitely many times as i runs over the odd integers bigger than 1. We call these numbers the limit points of Nathanson’s lambda sequences.

Satyanand Singh

Title: Combinatorial and Additive Number Theory IV
Editor: Melvyn B. Nathanson
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-67996-5
Print ISBN: 978-3-030-67995-8
DOI: https://doi.org/10.1007/978-3-030-67996-5

Springer Professional

Combinatorial and Additive Number Theory IV

CANT, New York, USA, 2019 and 2020

About this book

Table of Contents

Frontmatter

Extremal Sequences for Some Weighted Zero-Sum Constants for Cyclic Groups

On a Zero-Sum Problem Arising From Factorization Theory

Conditional Bounds on Siegel Zeros

Infinite Co-minimal Pairs in the Integers and Integral Lattices

Rigidity, Graphs and Hausdorff Dimension

On Generalized Harmonic Numbers

Partitions for Semi-magic Squares of Size Three

A Sum of Negative Degrees of the Gaps Values in Two-Generated Numerical Semigroups and Identities for the Hurwitz Zeta Function

Widely Digitally Stable Numbers

Non-injectivity of Nonzero Discriminant Polynomials and Applications to Packing Polynomials

Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

Bounds on Point Configurations Determined by Distances and Dot Products

Distribution of Missing Differences in Diffsets

Recent Progress in Hilbert Cubes Theory

Intrinsic Characterization of Representation Functions and Generalizations

Combinatorics of Multicompositions

On the Connection Between the Goldbach Conjecture and the Elliott-Halberstam Conjecture

Part-Frequency Matrices, II: Recent Work

A Conjectural Inequality for Visible Points in Lattice Parallelograms

On a Two-Dimensional Exponential Sum

On Consecutive Perfect Powers with Elementary Methods

Sidon Sets and Perturbations

Multiplicative Representations of Integers and Ramsey’s Theorem

On Distinct Consecutive Differences

Limit Points of Nathanson’s Lambda Sequences

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