Correct order on some certain weighted representation functions
Year: 2024
Type: article
Abstract: Let ℕ be the set of all nonnegative integers. For any positive integer k and any subset A of nonnegative integers, let r 1 , k ( A , n ) be the number of solutions ( a 1 , a 2 ) to the equation n = a 1 + k a 2 . In 2016, Qu proved that lim inf n → ∞ r 1 , k ( A , n ) = ∞ providing that r 1 , k ( A , n ) = r 1 , k ( ℕ ∖ A , n ) for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu’s result and obtained that lim inf n → ∞ r 1 , k ( A , n ) log n > 0 . In this note, we further improve the lower bound on r 1 , k ( A , n ) by showing that lim inf n → ∞ r
Source: Comptes Rendus Mathématique
DOI: 10.5802/crmath.573
Institutions: Anhui Normal University, Yangzhou University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4399216997