On a sumset problem of dilates

Sandeep Singh Chahal () and Ram Krishna Pandey ()
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Sandeep Singh Chahal: Akal University
Ram Krishna Pandey: Indian Institute of Technology Roorkee

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 4, 1180-1185

Abstract: Abstract Let A be a nonempty finite set of integers. For a real number m, the set $$m\cdot A=\{ma: a\in A\}$$ m · A = { m a : a ∈ A } denotes the set of m-dilates of A. In 2008, Bukh initiated an interesting problem of finding a lower bound for the sumset of dilated sets, i.e., a lower bound for $$|m_1\cdot A+m_2\cdot A+ \cdots +m_n\cdot A|$$ | m 1 · A + m 2 · A + ⋯ + m n · A | , where $$m_1, m_2, \ldots , m_n$$ m 1 , m 2 , … , m n are integers. In this paper, we consider the case $$|3\cdot A+k\cdot A|$$ | 3 · A + k · A | , where k is a prime number $$\ge 5$$ ≥ 5 . Under some assumptions on A, first we give a general lower bound on the cardinality of $$3\cdot A+k\cdot A$$ 3 · A + k · A then, for large sets A, under the same assumptions, we improve this general lower bound. The results also hold true for the general sumset $$q \cdot A + k\cdot A$$ q · A + k · A , where q is any odd prime $$

Keywords: Additive combinatorics; Sumsets of dilates; 11B13; 11P70 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00089-6

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