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Number Theoretic Density and Logical Limit Laws
Stanley N. Burris, University of Waterloo, ON, Canada
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Mathematical Surveys and Monographs
2001; 289 pp; hardcover
Volume: 86
ISBN-10: 0-8218-2666-2
ISBN-13: 978-0-8218-2666-9
List Price: US$80
Member Price: US$64
Order Code: SURV/86
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This book shows how a study of generating series (power series in the additive case and Dirichlet series in the multiplicative case), combined with structure theorems for the finite models of a sentence, lead to general and powerful results on limit laws, including \(0 - 1\) laws. The book is unique in its approach to giving a combined treatment of topics from additive as well as from multiplicative number theory, in the setting of abstract number systems, emphasizing the remarkable parallels in the two subjects. Much evidence is collected to support the thesis that local results in additive systems lift to global results in multiplicative systems.

All necessary material is given to understand thoroughly the method of Compton for proving logical limit laws, including a full treatment of Ehrenfeucht-Fraissé games, the Feferman-Vaught Theorem, and Skolem's quantifier elimination for finite Boolean algebras. An intriguing aspect of the book is to see so many interesting tools from elementary mathematics pull together to answer the question: What is the probability that a randomly chosen structure has a given property? Prerequisites are undergraduate analysis and some exposure to abstract systems.

Readership

Graduate students and research mathematicians interested in combinatorics, number theory and logic.

Reviews

"Shows an exciting connection between combinatorics, number theory and logic, and certainly deserves to be more widely known. The book gives a very clear account of it and it is easily readable."

-- Zentralblatt MATH

"This book is a lucid, self-contained introduction to a fascinating interaction between analysis, combinatorics, logic and number theory ... accessible to an undergraduate and gives interesting examples to illustrate the concepts."

-- Mathematical Reviews

Table of Contents

Additive number systems
  • Background from analysis
  • Counting functions, fundamental identities
  • Density and partition sets
  • The case \(\rho = 1\)
  • The case \(0 < \rho < 1\)
  • Monadic second-order limit laws
Multiplicative number systems
  • Background from analysis
  • Counting functions and fundamental identities
  • Density and partition sets
  • The case \(\alpha = 0\)
  • The case \(0 < \alpha < \infty\)
  • First-order limit laws
  • Appendix A. Formal power series
  • Appendix B. Refined counting
  • Appendix C. Consequences of \(\delta(\mathsf P) = 0\)
  • Appendix D. On the monotonicity of \(a(n)\) when \(p(n) \leq 1\)
  • Appendix E. Results of Woods
  • Bibliography
  • Symbol index
  • Subject index
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