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Journal of the Ramanujan Mathematical Society

Volume 36, Issue 3, September 2021  pp. 179–191.

On a problem in additive number theory

Authors:  Mohan Chintamani, Shanta Laishram and Prabal Paul
Author institution:School of Mathematics and Statistics, University of Hyderabad, Hyderabad, India

Summary:  Let A be a non-empty subset of a finite abelian group G. For x ∈ G, let r{A - A} (x) = # {(a,a') ∈ A × A: x = a - a'} the number of representations of x as a difference of two elements from A. Lev \cite{Lev} proposed the following problem: if r{A-A}(x) ≥ {|A|}/{2}, ∀ x ∈ A - A, is it necessarily true that A - A is either a subgroup or a union of three cosets of a subgroup? By an example, we illustrate that the problem has negative answer for a non cyclic group G. We give an affirmative answer to this problem for a large class of subsets A of a cyclic group G.

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