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This impressive volume is dedicated to Mel Nathanson, a leading authoritative expert for several decades in the area of combinatorial and additive number theory. Nathanson's numerous results have been widely published in top notch journals and in a number of excellent graduate textbooks (GTM Springer) and reference works.For several decades, Mel Nathanson's seminal ideas and results in combinatorial and additive number theory have influenced graduate students and researchers alike. The invited survey articles in this volume reflect the work of distinguished mathematicians in number theory, and represent a wide range of important topics in current research.
Sommario
Addictive Number Theory.- Sum-Product Theorems and Applications.- Can You Hear the Shape of a Beatty Sequence?.- Variance of Signals and Their Finite Fourier Transforms.- Sparse Sets in Time and Frequency Related to Diophantine Problems and Integrable Systems.- Addition Theorems in Acyclic Semigroups.- Small Sumsets in Free Products of $$\mathbb{Z}/2\mathbb{Z}$$ .- A Combinatorial Approach to Sums of Two Squares and Related Problems.- A Note on Elkin's Improvement of Behrend's Construction.- Distinct Matroid Base Weights and Additive Theory.- The Postage Stamp Problem and Essential Subsets in Integer Bases.- A Universal Stein-Tomas Restriction Estimate for Measures in Three Dimensions.- On the Exact Order of Asymptotic Bases and Bases for Finite Cyclic Groups.- The Erd?s-Turán Problem in Infinite Groups.- A Tiling Problem and the Frobenius Number.- Sumsets and the Convex Hull.- Explicit Constructions of Infinite Families of MSTD Sets.- An Inverse Problem in Number Theory and Geometric Group Theory.- Cassels Bases.- Asymptotics of Weighted Lattice Point Counts Inside Dilating Polygons.- Support Bases of Solutions of a Functional Equation Arising From Multiplication of Quantum Integers and the Twin Primes Conjecture.- Exponential Sums and Distinct Points on Arcs.- New Vacca-Type Rational Series for Euler's Constant ? and Its "Alternating" Analog $$\ln \frac{4}{\pi }$$ .- Mixed Sums of Primes and Other Terms.- Classes of Permutation Polynomials Based on Cyclotomy and an Additive Analogue.
Riassunto
Providing an up-to-date snapshot of the field, this volume presents a broad introduction to the current state of additive number theory. The applications discussed include communication theory and digital signal processing, image processing, optimization and discrete geometry.
