On Nathanson's Triangular Number Phenomenon
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912540509863936 |
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| author | O'Bryant, Kevin |
| author_facet | O'Bryant, Kevin |
| contents | For a finite set $A\subseteq \mathbb{Z}$, the $h$-fold sumset is $hA :=\{x_1+\dots+x_h:x_i\in A\}$. We interpret the beginning of the sequence of sumset sizes $(|hA|)_{h=1}^\infty$ in terms of the successive $L^1$-minima of a lattice (specifically, the points in $\mathbb{Z}^{|A|}$ whose coordinates sum to 0 and which are perpendicular to $\langle a_1,\dots,a_{|A|}\rangle$). In particular, if $h_1,h_2$ are the first and second minima, and $1\le h<h_1$, then $|hA|=\binom{h+|A|-1}{|A|-1}$, while if $h_1\le h <h_2$, then $|hA|=\binom{h+|A|-1}{|A|-1}-\binom{h-h_1+|A|-1}{|A|-1}$. This explains the appearance of triangular numbers in the sequence of sumset sizes, an observation related to a recent experiment of Nathanson. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Nathanson's Triangular Number Phenomenon O'Bryant, Kevin Number Theory Combinatorics 11B05, 11B13, 11B30, 11B34, 11B75 For a finite set $A\subseteq \mathbb{Z}$, the $h$-fold sumset is $hA :=\{x_1+\dots+x_h:x_i\in A\}$. We interpret the beginning of the sequence of sumset sizes $(|hA|)_{h=1}^\infty$ in terms of the successive $L^1$-minima of a lattice (specifically, the points in $\mathbb{Z}^{|A|}$ whose coordinates sum to 0 and which are perpendicular to $\langle a_1,\dots,a_{|A|}\rangle$). In particular, if $h_1,h_2$ are the first and second minima, and $1\le h<h_1$, then $|hA|=\binom{h+|A|-1}{|A|-1}$, while if $h_1\le h <h_2$, then $|hA|=\binom{h+|A|-1}{|A|-1}-\binom{h-h_1+|A|-1}{|A|-1}$. This explains the appearance of triangular numbers in the sequence of sumset sizes, an observation related to a recent experiment of Nathanson. |
| title | On Nathanson's Triangular Number Phenomenon |
| topic | Number Theory Combinatorics 11B05, 11B13, 11B30, 11B34, 11B75 |
| url | https://arxiv.org/abs/2506.20836 |