Given n non-zero real numbers v_1,…,v_n, what is the maximum possible number of subsets {i_1,…,i_k} that can have the same subset sum v_{i_1}+…+v_{i_n}? Littlewood and Offord raised the question, showed that the answer is at most 2^nlog(n)/\sqrt{n} and conjectured that the log(n) can be removed. Erdos proved the conjecture and raised more questions that have continued to attract attention, primarily relating the arithmetic structure of v_1,…,v_n and the maximum number of subsets with a given subset-sum.

In the first lecture, we shall review and prove several of these results (due to Littlewood–Offord, Erdos, Moser, Stanley, Halasz, Tao–Vu, Rudelson–Vershynin…) and show an application to random matrices.

In the second part, we start with a question in random polynomials and see how it leads to a variant of the Littlewood–Offord problem that has not been considered before.

Most of the material presented should be accessible to undergraduate students. Much of the lecture is an outcome of my joint study in summer with Sourav Sarkar (ISI, Kolkata).

- All seminars.
- Seminars for 2014