In this third part of our work, we go back to the study of the $\nu^{(k)}_G$ functions (introduced in the first one), which count the minimal cardinality of a sumset containing an element with a single representation. An upper bound for these functions is obtained in the case $k=2$ using what we call the generalized increasingly small sumsets property, which is proved to hold for all Abelian groups. Moreover, we show that our bound cannot be improved in general.
Publié le : 2007-09-15
Classification:
additive number theory,
small sumsets,
supersmall sumsets,
Abelian groups,
initial segment,
11B75,
20D60,
11P99,
20Kxx
@article{1229619661,
author = {Plagne, Alain},
title = {Optimally small sumsets in groups III. The generalized increasingly small sumsets property and the $\nu^{(k)}\_G$} functions},
journal = {Funct. Approx. Comment. Math.},
volume = {37},
number = {1},
year = {2007},
pages = { 377-397},
language = {en},
url = {http://dml.mathdoc.fr/item/1229619661}
}
Plagne, Alain. Optimally small sumsets in groups III. The generalized increasingly small sumsets property and the $\nu^{(k)}_G$} functions. Funct. Approx. Comment. Math., Tome 37 (2007) no. 1, pp. 377-397. http://gdmltest.u-ga.fr/item/1229619661/