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A note on representation functions with different weights

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Qu Z.

Colloquium Mathematicum

2016

tom 143

nr 1

105-112

For any positive integer k and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1,k}(A,n) = r_{1,k}(ℕ ∖ A,n)$ and $r_{1,l}(A,n) = r_{1,l}(ℕ ∖ A,n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1,k}(A,n) = r_{1,k}(ℕ ∖ A,n)$ for all n ≥ n₀, we have $r_{1,k}(A,n) → ∞$ as n → ∞.

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1 Colloquium Mathematicum

1 Qu Z.

1 2016

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A note on representation functions with different weights