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Diversity, equity, and inclusion for problems in additive number theory

Melvyn B. Nathanson

Abstract

This is a survey of the diversity of problems in additive number theory. Equity requires the consideration of less currently popular problems, and suggests their inclusion in the additive canon. Of particular interest are problems about the sizes of sumsets of finite sets of integers and problems about the arithmetical structure of intersections of sumsets.

Diversity, equity, and inclusion for problems in additive number theory

Abstract

This is a survey of the diversity of problems in additive number theory. Equity requires the consideration of less currently popular problems, and suggests their inclusion in the additive canon. Of particular interest are problems about the sizes of sumsets of finite sets of integers and problems about the arithmetical structure of intersections of sumsets.
Paper Structure (11 sections, 18 theorems, 52 equations)

This paper contains 11 sections, 18 theorems, 52 equations.

Table of Contents

  1. Canonical additive problems
  2. Sum-product: A problem no one cared about
  3. Freiman's theorem
  4. A noncanonical problem: Sizes of sumsets
  5. Range of sumset sizes
  6. Missing sumset sizes for 3-fold sums of sets with 3 elements
  7. Computational complexity
  8. Sumset races
  9. Lebesgue measurable sets
  10. Sumset intersection limits
  11. The moral of the story

Key Result

Theorem 1

For all positive integers $k$, let

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Nathanson nath25a
  • Theorem 5: Rajagopal raja25
  • Theorem 6: Nathanson nath25a
  • Theorem 7: Vincent Schinina schi25
  • Theorem 8: Rajagopal raja25
  • Theorem 9: Nathanson nath26a
  • Theorem 10: Kravitz krav25
  • ...and 8 more