## Some new results on sum-product inequalities, and a new method for attacking certain additive problems

#### Ernest CrootGeorgia Institute of Technology

In this talk I will discuss three papers that are joint with Derrick Hart and Olof Sisask (the first two are joint with Derrick, while the last is joint with Olof). The results with Derrick concern the problem of determining the size of kA, A a set of n real numbers, given that the product set A.A is small''. We establish new bounds on this problem, and also establish a weak Erdos-Szemeredi conjecutre'' for function fields; and, under certain number theoretical conjectures, this number field'' result also holds for the integers. The paper with Olof introduces a new method for attacking additive problems. I will discuss how we have used this to very recently (as in within the past three days) get new bounds on the length of the longest arithmetic progression in a sumset A+B; furthermore, the proof is completely combinatorial and elementary (no Fourier analysis, or Gowers uniformity).

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