Burgess Inequality In {\mathbb {F}_{p^2}}

The purpose of the paper is to present new estimates on incomplete character sums in finite fields that are of the strength of Burgess’ celebrated theorem for prime fields. More precisely, an inequality of this type is obtained in F_p²{F_{p^2}} and also for binary quadratic forms, improving on the work of Davenport–Lewis and on several results due to Burgess. The arguments are based on new estimates for the multiplicative energy of certain sets that allow us to improve the amplification step in Burgess’ method.