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- Title
On bounded basis with prescribed representation functions.
- Authors
Xue, Fang-Gang
- Abstract
Let ℤ be the set of integers and ℕ the set of positive integers. For a nonempty set A of integers and any integers n , h with h ≥ 2 , define r A , h (n) as the number of solutions of n = a 1 + ⋯ + a h , where a 1 ≤ ⋯ ≤ a h and a i ∈ A for i = 1 , ... , h. A set A of integers is defined as a basis of order h for ℤ if r A , h (n) ≥ 1 for every integer n. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded bases. In this paper, we generalize the above result and obtain that: for any function f : ℤ → ℕ , there exists a bounded basis of order h for ℤ such that r A , h (n) = f (n) for every integer n.
- Subjects
INTEGERS; LOGICAL prediction
- Publication
International Journal of Number Theory, 2024, Vol 20, Issue 2, p349
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S1793042124500179