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

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

• Additive number theory (Generalized Zeckendorf Decompositions): Prerequisites are basic probability and combinatorics. It turns out an alternative definition of the Fibonacci numbers is that they are the unique sequence of positive integers such that all positive integers can be uniquely written as sums of non-adjacent elements of the set. There are a lot of questions one can ask about such decompositions, from the distribution of number of summands (Gaussian) to the gaps between summands (geometric decay) to the largest gap (similar to coin tossing), .... The background for these projects is minimal.
  • Papers
    • On the number of summands in Zeckendorf decompositions
    • From Fibonacci Numbers to Central Limit Type Theorems
    • The Average Gap Distribution for Generalized Zeckendorf Decompositions
    • Gaussian Behavior in Generalized Zeckendorf Decompositions
    • Gaps between summands in generalized Zeckendorf decompositions
    • Generalizing Zeckendorf's Theorem to f-decompositions
  • Talks
    • Cookie Monster Meets the Fibonacci Numbers. Mmmmmm -- Theorems!: http://youtu.be/5e6HsfxqVSE (slides here).

We talked about a variety of projects, including 2-dimensional representations, rates of convergence, ....