Combinatorial and Additive Number Theory V: CANT, New York, USA, 2021 (Springer Proceedings in Mathematics & Statistics, 395)

This proceedings volume, the fifth in a series from the Combinatorial and Additive Number Theory (CANT) conferences, is based on talks from the 19th annual workshop, held online due to the COVID-19 pandemic. Organized every year since 2003 by the New York Number Theory Seminar at the CUNY Graduate Center, the workshops survey state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. The CANT 2021 meeting featured over a hundred speakers from North and South America, Europe, Asia, Australia, and New Zealand, and was the largest CANT conference in terms of the number of both lectures and participants. These proceedings contain peer-reviewed and edited papers on current topics in number theory. Topics featured in this volume include sumsets, minimal bases, Sidon sets, analytic and prime number theory, combinatorial and discrete geometry, numerical semigroups, and a survey of expansion, divisibility, and parity. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory. Preface: Mathematics in the Time of COVID Contents On the Number of Dot Product Chains in Finite Fields and Rings 1 Introduction Background Notation Main Results Sharpness and Relevance of Hypotheses 2 Bounds on 3-Chains in mathbbFqd 3 Bounds on k-Chains in mathbbZqd 4 Bounds on k-chains in mathbbFqd 5 Small Set Results Proof of Theorem 4 Erratum References Completeness of Positive Linear Recurrence Sequences 1 Introduction 2 Modifying Sequences Maximal Complete Sequence Modifications of Sequences of Arbitrary Coefficients 3 Families of Sequences Using 1s and 0s as Initial Coefficients The ``2L—1 Conjecture'' 4 An Analytical Approach An Introduction to Principal Roots Applications to Completeness Denseness of Incomplete Roots 5 Open Questions 6 Brown's Criterion and a Corollary 7 Lemmas for Sect.2 8 Lemmas for Sect.3 9 Lemmas for Sect.4 References Length Density and Numerical Semigroups 1 Introduction 2 Background 3 Asymptotics and Computation 4 Families of Numerical Semigroups Supersymmetric Numerical Semigroups Embedding Dimension 3 Numerical Semigroups Maximal Embedding Dimension Numerical Semigroups 5 Tasty and Bland Gluings of Numerical Semigroups References On a Problem of Cilleruelo and Nathanson, II 1 Introduction 2 Proof of Theorem 1 3 Proof of Theorem 2 References Linked Partition Ideals and a Schur-Type Identity of Andrews 1 Introduction 2 Linked Partition Ideals and a Matrix Equation 3 q-Borel Operators 4 Non-standard Generating Function 5 Theorem 1 6 The Continuous q-Hermite Polynomials References Semi-magic Matrices for Dihedral Groups 1 Introduction 2 Definitions and Notations 3 Structure of D2n 4 Irreducible Types for ρ 5 The Intertwining Operator ρ 6 Semi-magic Matrices 7 Semi-magic Squares for MM(D2n) 8 Alternative Counting Formulas 9 Orthogonal Idempotents for MM(D2n) 10 Quaternionic Bases References Is the Syracuse Falling Time Bounded by 12? 1 Introduction 2 Falling Time Records Integers Satisfying `3́9`42`"̇613A``45`47`"603Aft(n) > 14 Glide Records Falling Time Distribution A Variant of Jumps 3 The Syracuse Version Current Maximum A Variant of Syracuse Jumps 4 The Case 2ell-1 5 For Very Large n Heuristics A Challenge References Genera of Numerical Semigroups and Polynomial Identities for Degrees of Syzygies 1 Introduction 2 Polynomial Identities for Numerical Semigroups Polynomial Identities of Arbitrary Degree Derivatives Φ(r)z=1, Ψ(r)z=1 and Π(r) z=1 3 Linear Equations for Alternating Power Sums mathbbCk(Sm) 4 Algebraic Equations for Kr in Numerical Semigroups Numerical Semigroups S3 5 Supplementary Relations for Kr and Gr in Symmetric Semigroups Symmetric (Not CI) Numerical Semigroups S4 Supplementary Relations for Kr and Gr in Symmetric CI Semigroups 6 Concluding Remarks References Lp Estimates for Bilinear Generalized Radon Transforms in the Plane 1 Introduction 2 A General Class of Bilinear Generalized Radon Transforms 3 Sharpness Examples 4 Summary of Sharpness Conditions and Vertices of Q(Bθ) 5 Trivial Bounds 6 The L32 timesL32 toL1 Estimate for the Model Operators 7 The L2,1 timesL2,1 toL2 Estimate for the Model Operators, 0