Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis
| dc.creator | Farhi, Bakir | |
| dc.date | 2009-02-18 | |
| dc.date.accessioned | 2026-07-07T12:43:31Z | |
| dc.date.available | 2026-07-07T12:43:31Z | |
| dc.description | Let $A$ be an additive basis of order $h$ and $X$ be a finite nonempty subset of $A$ such that the set $A \setminus X$ is still a basis. In this article, we give several upper bounds for the order of $A \setminus X$ in function of the order $h$ of $A$ and some parameters related to $X$ and $A$. If the parameter in question is the cardinality of $X$, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in $h$ with degree $(|X| + 1)$. Here, by taking instead of the cardinality of $X$ the parameter defined by $d := \frac{\diam(X)}{\gcd\{x - y | x, y \in X\}}$, we show that the order of $A \setminus X$ is bounded above by $(\frac{h (h + 3)}{2} + d \frac{h (h - 1) (h + 4)}{6})$. As a consequence, we deduce that if $X$ is an arithmetic progression of length $\geq 3$, then the upper bounds of Nathanson and Nash are considerably improved. Further, by considering more complex parameters related to both $X$ and $A$, we get upper bounds which are polynomials in $h$ with degree only 2. | |
| dc.description | 17 pages | |
| dc.identifier | https://arxiv.org/abs/0902.3093 | |
| dc.identifier | http://arxiv.org/abs/0902.3093 | |
| dc.identifier | J. Number Theory, 128 (2008), p. 2214-2230 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/220445 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 11B13 | |
| dc.title | Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis | |
| dc.type | text |