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  • Difference bases in cyclic groups
    Banakh, Taras, 1968- ; Gavrylkiv, Volodymyr
    ▫$A$▫ subset ▫$B$▫ of an Abelian group ▫$G$▫ is called a difference basis of ▫$G$▫ if each element ▫$g \in G$▫ can be written as the difference ▫$g = a-b$▫ of some elements ▫$a,b \in B$▫. The ... smallest cardinality ▫$|B|$▫ of a difference basis ▫$B \subset G$▫ is called the difference size of ▫$G$▫ and is denoted by ▫$\Delta[G]$▫. We prove that for every ▫$n \in \mathbb{N}$▫ the cyclic group ▫$C_n$▫ of order ▫$n$▫ has difference size ▫$\frac{1+\sqrt{4|n|-3}}{2} \le \Delta[C_n] \le \frac{3}{2} \sqrt{n}$▫. If ▫$n \ge 9$▫ (and ▫$n \ge 2 \cdot 10^{15}$▫), then ▫$\Delta[C_n] \le \frac{12}{\sqrt{73}} \sqrt{n}$▫ (and ▫$\Delta[C_n] < \frac{2}{\sqrt{3}} \sqrt{n})$▫. Also, we calculate the difference sizes of all cyclic groups of cardinality ▫$\le 100$▫.
    Source: Journal of algebra and its applications. - ISSN 0219-4988 (Vol. 18, no. 5, 2019, 1950081 [18 str.])
    Type of material - article, component part
    Publish date - 2019
    Language - english
    COBISS.SI-ID - 18403161

    Link(s):

    https://www.worldscientific.com/doi/abs/10.1142/S0219498819500816
    DOI

source: Journal of algebra and its applications. - ISSN 0219-4988 (Vol. 18, no. 5, 2019, 1950081 [18 str.])

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