Honors Oral Exam

Probabilistic and Experimental Methods in Sum-Product Theory.

Xiaobo Luo (University of Rochester)

Friday, April 23rd, 2021
8:45 AM - 9:45 AM
Zoom Meeting ID 986 2892 7517
Sum-Product Theory has been analyzed extensively in the past few decades. In particular, given a subset \(A\) of a ring, researchers look for lower bounds on $$max( A+A , AA )\(, the larger of the size of the sum and product sets of\)A$$.

In this paper, I investigate the setting where A is a random subset of \(\{1,\dots,N \}\) or \(Z_N\) where each element is chosen to lie in the subset with probability \(p\). I come up with a closed-form expression for the expected size of \(A+A\) in two scenarios. The expected size of \(A*A\) is also calculated and a closed-form lower bound is obtained. We approach this problem using both the probabilistic method and experimental methods which can aid us in gaining intuition and in checking our results.

Event contact: jonathan dot pakianathan at rochester dot edu