|
Dr. Kevin O'Bryant I enjoy applications of analysis and probability to number theory and combinatorics. Additive number theory is one area with many examples of this: let A be a subset of {1, 2, ..., n} and let S be the set of numbers that are the sum of two elements of A; if |S| > |A|2 / 4, how large can |A| be? The best partial solutions to this problem use real analysis and probability. Another example is Fraenkel's Tiling Conjecture: suppose that each integer is in exactly one of the sets B(xi,yi)={ [n xi + yi] }, where [z] is the largest integer not larger than z, i = 1..m, m > 2, xi and yi are real numbers, and n varies through the integers; then there are i < j with xi = xj. A main tool used recently to attack the still-open conjecture is Fourier analysis. |