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Sum-Product and Character Sums in finite fields

In this talk I will present estimates on incomplete character sums in finite fields, with special emphasize on the non-prime case. Some of the results are of the same strength as Burgess celebrated theorem for prime fields. The improvements are mainly based on arguments from arithmetic combinatorics providing new bounds on multiplicative energy and an improved amplification strategy. In particular, we improve on earlier work of Davenport-Lewis and Karacuba.