On the maximum values of the additive representation functions

Kiss Z., Sándor and Sándor, Csaba (2016) On the maximum values of the additive representation functions. International Journal of Number Theory, 12 (4). pp. 1055-1075. ISSN 1793-0421

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Abstract

Let A and B be sets of nonnegative integers. For a positive integer n let R_A(n) denote the number of representations of n as the sum of two terms from A. Let s_A(x) = max_{n\le x}R_A(n) and d_{A;B}(x) = max_{ t: a_t \le x or b_t \le x}|a_t-b_t|. In this paper we study the connection between s_A(x), s_B(x) and d_{A;B}(x). We improve a result of Haddad and Helou about the Erdős - Turán conjecture.

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