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.
SUMSETS AND CARRIES IN CYCLIC GROUPS / F. Monopoli ; tutor: G. Molteni, A. Perelli; coordinatore: L. van Geemen. DIPARTIMENTO DI MATEMATICA "FEDERIGO ENRIQUES", 2015 Dec 14. 28. ciclo, Anno Accademico 2015. [10.13130/monopoli-francesco_phd2015-12-14].
SUMSETS AND CARRIES IN CYCLIC GROUPS
F. Monopoli
2015
Abstract
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.File | Dimensione | Formato | |
---|---|---|---|
phd_unimi_R09950.pdf
accesso aperto
Tipologia:
Tesi di dottorato completa
Dimensione
739.66 kB
Formato
Adobe PDF
|
739.66 kB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.