Combinatorial and Additive Number Theory IV: CANT, New York, USA, 2019 and 2020 (Springer Proceedings in Mathematics & Statistics, 347)

"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"--Back cover Preface: Math in the Time of Cholera Contents Extremal Sequences for Some Weighted Zero-Sum Constants for Cyclic Groups 1 Introduction 2 Some Preliminaries 3 The Case A=mathbbZn{0} 4 The Case Where A is the Set of Quadratic Residues Modulo p 5 The Case A=mathbbZn* References On a Zero-Sum Problem Arising From Factorization Theory 1 Introduction 2 Background on Sets of Lengths 3 Proof of Theorems 1 and 2 References Conditional Bounds on Siegel Zeros 1 Introduction 2 Background 2.1 Repulsion Property 2.2 Bounds for L(1,χ) 3 Better Siegel Zero Bounds from Weak Goldbach Conjectures 3.1 Questions References Infinite Co-minimal Pairs in the Integers and Integral Lattices 1 Introduction 1.1 Statement of Results 2 A Co-minimal Pair Involving a Bounded Below Subset 3 A Co-minimal Pair Involving an Infinite Symmetric Subset 4 Co-minimal Pairs in the Free Abelian Groups of Higher Rank References Rigidity, Graphs and Hausdorff Dimension 1 Introduction 2 Main Result 3 Definitions and Statements of Distance-Type Results 3.1 Notation 3.2 Configurations, Frameworks and Distances 3.3 Independence and Genericity 3.4 Structural Rigidity 3.5 Statements of Distance-Type Results 4 The Lebesgue Measure on the Reduced Moduli Space of Congruence Classes 4.1 Step 1: Passage to Origin Pinned Configurations 4.2 Step 2: Analysis of the Orthogonal Group Action on Pinned Configurations and ``Moving Frames'' 4.3 Step 3: Quotienting the Action of the Orthogonal Group 4.4 Deduction of the Main Theorem 5 Graph Distances of Subsets of Rd 5.1 Examples of Distance Sets 5.2 A Sharp Upper Bound for the Dimension of the Distance Set 5.3 Bounds on the Number of Noncongruent Realizations 5.4 The Proof of the Dimensional Threshold 5.5 The Natural Measure nu g on E-gE 6 Geometric Results 6.1 Generic Frameworks 6.2 Useful Lemmas 7 Proof of Theorem 3.26 References On Generalized Harmonic Numbers 1 Introduction 2 Preliminary Lemmas 3 Proof of Theorem 1 4 Proof of Theorem 2 4.1 The Estimates for Δ1 and Δ'1 4.2 The Estimates for Δ2, Δ'2, and Δ3 4.3 The Estimates for Δ4 and Δ5 References Partitions for Semi-magic Squares of Size Three 1 Introduction 2 Magic Squares and Clebsch-Gordan Coefficients 3 A Partition for Semi-magic Squares of Size Three 4 Examples 5 Examples of Zeros 6 Orbits and a Second Partition 7 Trivial Zeros 8 Triangles for Trivial Zeros 9 Example: Triangles for J=15 References A Sum of Negative Degrees of the Gaps Values in Two-Generated Numerical Semigroups and Identities for the Hurwitz Zeta Function 1 Introduction 2 A Sum of the Inverse Gaps Values g-1(S2) 3 A Sum of the Negative Degrees of Gaps Values g-n(S2) 4 An Application to the Hurwitz Zeta Function References Widely Digitally Stable Numbers 1 Introduction 2 Initial Ideas Behind the Proof 3 The Coverings for the Leading Zeros 4 The Coverings for the Leading Sevens 5 The Right-Most Digits References Non-injectivity of Nonzero Discriminant Polynomials and Applications to Packing Polynomials 1 Background 2 Quadratic Packing Polynomials on Sectors 3 Non-injectivity When Discriminant Is Zero 4 Applications to Packing Polynomials References Representing Sequence Subsums as Sumsets of Near Equal Sized Sets 1 Introduction 2 Partitioning Results for General n 3 Partitioning Results for Large n References Bounds on Point Configurations Determined by Distances and Dot Products 1 Introduction 1.1 Background 1.2 Main Results 1.3 Organization of This Paper 2 Preliminaries 2.1 The α-Line for a Point p 2.2 Sketch of Proof of Theorem 3 2.3 The Szemerédi–Trotter Theorem 3 Proofs 3.1 Proof of Theorem 8 3.2 Proof of Theorem 9 3.3 Proofs of Theorems 10 and 11 3.4 Proof of Theorem 12 3.5 Proof of Corollary 2 3.6 Proof of Theorem 14 3.7 Proof of Theorem 15 3.8 Proof of Theorem 16 References Distribution of Missing Differences in Diffsets 1 Introduction 1.1 Background 1.2 Distribution of |S-S| when n=35 1.3 Main Results 2 Results about Having (Few) Differences 3 Results about Missing (Few) Differences 3.1 Intuitively Measuring the Limiting Probabilities 3.2 Using Conditional Probabilities 3.3 Calculations and Results 3.4 About Rulers 4 Conjectures 5 Distribution of |S-S| when nle36 6 Code for Estimating j(2k) References Recent Progress in Hilbert Cubes Theory 1 Introduction 2 Notation 3 Some Previous Results 4 Proofs References Intrinsic Characterization of Representation Functions and Generalizations 1 Introduction 2 Characterization of Binary Representation Functions 3 Counting and Characteristic Functions for Binary Sums 4 Representation Functions for m-ary Sums References Combinatorics of Multicompositions 1 Introduction 2 Multicompositions and Generating Functions 3 Counting Multicompositions by Various Parts 3.1 Number of All Parts 3.2 Number of Positive Parts 3.3 Number of Zeros 4 Restricted Part Multicompositions 5 Connections to Diagonal Sums of Triangles 6 Exclusion Statistics and Further Work References On the Connection Between the Goldbach Conjecture and the Elliott-Halberstam Conjecture 1 Introduction 2 Preliminary Lemmata 3 Proof of Theorem 1 and Corollary 1 3.1 Evaluation of S1(α) Using EH(Nθ (logN)2A + 8) 3.2 Evaluation of S2(α) Using EHμ(N1 - θ) 3.3 Proof of Theorem 1 3.4 Proof of Corollary 1 References Part-Frequency Matrices, II: Recent Work 1 Introduction 2 On m-Ary Partitions 3 Partitions into Two or Three Part Sizes 4 Partitions with Parts Separated by Parity 5 Investigations References A Conjectural Inequality for Visible Points in Lattice Parallelograms 1 Introduction and Notation 2 A Reduction Result 2.1 Motivation for Studying This Problem 3 V(a,n)=V(a-1 12mumodn,n) 4 Counting Visible Points in Pa,n 5 Some Numerics 6 Further Speculative Remarks References On a Two-Dimensional Exponential Sum 1 Introduction 2 Preliminaries 3 Proof of Theorem 1 4 Proof of Theorem 2 References On Consecutive Perfect Powers with Elementary Methods 1 Introduction 1.1 Notation 2 Proof of Theorem 1(i): q=2 3 Proof of Theorem 1(ii): p=2 3.1 Case q=3. 3.2 Case q ge5. 4 Proof of Theorem 1(iii): x is a power of 2 5 Proof of Theorem 1(iv): x equiv3,5 or 78mu(mod6mu8) 6 Proof of Theorem 1(v): x divides q 7 Proof of Theorem 1(vi): xequiv18mu(mod6muy) 8 Proof of Theorem 1(vii): y is Power of a Prime 9 Proof of Theorem 1(viii): ylemin((pq)pq,(qsqrtp2)p) 10 Closing Remarks References Sidon Sets and Perturbations 1 Sidon Sets 2 Perturbations of Countably Infinite Sets 3 h-Sidon Sets of Vectors in Fn 4 Open Problems References Multiplicative Representations of Integers and Ramsey's Theorem 1 Does One Solution Imply Many Solutions? 2 Additive Bases 3 Multiplicative Bases and Multiplicative Systems 4 An Iterated Ramsey's Theorem and Multiplicative Bases 5 A Doubly Iterated Ramsey's Theorem 6 Representation Functions of Multiplicative Systems 7 Open Problems References On Distinct Consecutive Differences 1 Introduction 2 Distinct Consecutive Differences 2.1 Proof of Theorem 1 2.2 Distinct Pairs of Consecutive Differences 3 A Construction for the Lower Bound 4 Convex Sets and |A+A -A| 5 Difference Sets of Convex Sets References Limit Points of Nathanson's Lambda Sequences 1 Introduction 1.1 Generating Special 2-Adic Representations 1.2 Generating Special g-Adic Representations for Odd g>1 2 Computing λ2,n(h) for hin{1,2,3} and Various Odd n>1. 3 Additional Results and Open Problems References