A new upper bound for the additive energy of the Heilbronn subgroup is found. Several applications to the distribution of Fermat quotients are obtained.
Using Stepanov’s method, we obtain an upper bound for the cardinality of the intersection of additive shifts of several multiplicative subgroups of a finite field. The resulting inequality is applied to a question dealing with the additive decomposability of subgroups.
This is the second paper of the series, started by Braverman-Finkelberg (2010) which describes a conjectural analogue of the affine Grassmannian for affine Kac-Moody groups (also known as double affine Grassmannian). The current paper is dedicated to describing a conjectural analogue of the convolution diagram for the double affine Grassmannian. In case our group is the special linear group, our conjectures can be derived from the work of Nakajima.