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Title of Journal: Sci China Math

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Abbravation: Science China Mathematics

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SP Science China Press

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DOI

10.1016/0005-7967(90)90034-g

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1869-1862

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On the values of representation functions

Authors: YongGao Chen
Publish Date: 2011/06/23
Volume: 54, Issue: 7, Pages: 1317-1331
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Abstract

For a set A of nonnegative integers the representation functions R 2A n and R 3A n are defined as the numbers of solutions to the equation n = a + a′ with a a′ ∈ A a a′ and a ⩽ a′ respectively Let ℕ be the set of nonnegative integers Given n 0 0 it is known that there exist A A′ ⊆ ℕ such that R 2A′ n = R 2ℕ A′ n and R 3A n = R 3ℕ A n for all n ⩾ n 0 We obtain several related results For example we prove that If A ⊆ ℕ such that R 3A n = R 3ℕ A n for all n ⩾ n 0 then 1 for any n ⩾ n 0 we have R 3A n = R 3ℕ A n c 1 n − c 2 where c 1 c 2 are two positive constants depending only on n 0 2 for any alpha frac1 16 the set of integers n with R 3A n αn has the density one The answers to the four problems in ChenTang 2009 are affirmative We also pose two open problems for further research

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