On the values of representation functions

Abstract

For a set A of nonnegative integers, the representation functions R 2 (A, n) and R 3 (A, 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. Givenfor all n n 0 . We obtain several related results. For example, we prove that: If A ⊆ N such that R 3 (A, n) = R 3 (N \ A, n) for all n n 0 , then (1) for any n n 0 we have R 3 (A, n) = R 3 (N \ A, n) > c 1 n − c 2 , where c 1 , c 2 are two positive constants depending only on n 0 ; (2) for any α < 1 16 , the set of integers n with R 3 (A, n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.

Partitions of natural numbers with the same weighted representation functions

“…Currently we have no results for Problem 1 with t 3. Motivated by Chen and Tang [3], and Chen [1], we pose the following problem.…”

“…If a 1 + ka 2 = k, then, either a 1 = k, a 2 = 0 or a 1 = 0, a 2 = 1. In the case n = k, (4) is equivalent to 2 = χ (k) + χ (0) + χ (0) + χ (1). Since χ (0) = 1 and χ (1) = 0, (4) is…”

On the nonvanishing of representation functions of some special sequences

“…In particular, Chen and Tang [2] proved the following theorems. The reader is referred to [1] for more results. …”

Partitions of $\mathbf{Z}_m$ with the Same Weighted Representation Functions

“…For i = 2, 3, Lev [6], Sándor [13] and Tang [14] determined all subsets A ⊂ N such that R i (A, n) = R i (N\A, n) for all n 2N − 1. The asymptotic behavior of the representation functions of these special sequences was studied by Chen and Tang (see [1,2]).…”