Course title:               Math 21-499: Introduction to Undergraduate Research

Instructor:                 Steven J. Miller (sjm1@williams.edu)

Additive number theory (MSTD sets): Prerequisites are basic probability and combinatorics. The questions involve the structure of sets with certain properties, higher dimensional generalizations, combinations of more than two sets, .... The background for these projects is minimal.

• Papers
  • Nathanson: Problems in additive number theory: great introduction to the subject and problems.
  • Martin-O'Bryant: Many sets have more sums than differences: first paper to prove a positive percentage of sets are sum-dominated (proof if probabilistic).
  • Hegarty-Miller: When almost all sets are difference domianted: almost all sets are difference dominated if we change the binomial model.
  • Miller-Orosz-Scheinerman: Constructing MSTD sets: elementary construction of a very big family of sum-dominated (or MSTD) sets.
  • Geoffrey Iyer, Oleg Lazarev, Steven J. Miller and Liyang Zhang: Generalized More Sums Than Differences Sets (shorter survey here).
  • Thao Do, Archit Kulkarni, Steven J Miller, David Moon and Jake Wellens: Sums and Differences of Correlated Random Sets
• Talks
  • Introduction to More Sums Than Difference Sets (7 minutes): https://youtu.be/Ji5UTWZ-Ioc
  • More Sum Than Difference Sets: http://youtu.be/gub8TS11Fr4 (Slides: slides for MSTD set lecture.) (40 minute talk)

We talked about several problems, especially the blip and phase transition in A+A+A vs A+A-A and finally about subsets of special sets (can there be an MSTD subset all of whose elements are prime)?

• Preprint of the phase transition in A+A+A here: http://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/MSTDPhaseTrnasition72.pdf
• For the divot: Distribution of Missing Sums in Sumsets (with Oleg Lazarev and Kevin O'Bryant), Experimental Mathematics (22 (2013), no. 2, 132--156).  pdf (see page 4)