A note on representation functions with different weights

Zhenhua Qu

Colloquium Mathematicae, Tome 144 (2016), p. 105-112 / Harvested from The Polish Digital Mathematics Library

Résumé

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 → ∞.

Détail

Publié le : 2016-01-01

Zbl 06545379

EUDML-ID : urn:eudml:doc:283601