SUMSETS AND CARRIES IN CYCLIC GROUPS

This thesis studies some direct and inverse problems concerning sumsets in cyclic groups, with applications to the study of carries. First of all we prove a generalization of the classical Cauchy-Davenport inequality in finite cyclic groups, thus bounding the cardinality of sumsets of sets satisfying certain density properties by a function linked to the arithmetic progression structure of the summands. This allows us to prove that digital sets modulo a generic integer which induce a minimal amount of distinct carries must be arithmetic progressions. After proving an inverse theorem for Pollard's inequality for sets with the Chowla property, we prove moreover that digital sets always induce carries with frequency asymptotically greater than 1/4. The last part of the thesis is devoted to the study of generalized sumsets, and contains various direct and inverse theorems for these objects in different ambient groups.