On monotone increasing representation functions
| Autor: | Kiss, Sándor Z., Sándor, Csaba, Yang, Quan-Hui |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$ denote the counting function of $A$.Bell and Shallit recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{\frac{k-2}{k}-\epsilon})$ for some $\epsilon>0$, then $R_{\mathbb{N}\setminus A,k}(n)$ is eventually strictly increasing. In this paper, we improve this result to $A(n)=O(n^{\frac{k-2}{k-1}})$. We also give an example to show that this bound is best possible. Comment: 11 pages |
| Databáze: | arXiv |
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