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Additive bases via Fourier analysis

Part of: Algebraic combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  29 April 2021

Bodan Arsovski
Bodan Arsovski*
Affiliation:
Skopje, North Macedonia
*
Email: [email protected]
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Abstract

Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

MSC classification

Primary: 11P99: None of the above, but in this section
Secondary: 05E99: None of the above, but in this section
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 6 , November 2021 , pp. 930 - 941
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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