Functiones et Approximatio Commentarii Mathematici

Optimally small sumsets in groups III. The generalized increasingly small sumsets property and the $\nu^{(k)}_G$} functions

Alain Plagne

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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.

Article information

Funct. Approx. Comment. Math., Volume 37, Number 2 (2007), 377-397.

First available in Project Euclid: 18 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B75: Other combinatorial number theory
Secondary: 20D60: Arithmetic and combinatorial problems 11P99: None of the above, but in this section 20Kxx: Abelian groups

additive number theory small sumsets supersmall sumsets Abelian groups initial segment


Plagne, Alain. Optimally small sumsets in groups III. The generalized increasingly small sumsets property and the $\nu^{(k)}_G$} functions. Funct. Approx. Comment. Math. 37 (2007), no. 2, 377--397. doi:10.7169/facm/1229619661.

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