Additive relations in irrational powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909942054649856 |
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| author | Harrison, Joseph |
| author_facet | Harrison, Joseph |
| contents | We investigate the additive theory of the set $S = \{1^c, 2^c, \dots, N^c\}$ when $c$ is a real number. In the language of additive combinatorics, we determine the asymptotic behaviour of the additive energy of $S$. When $c$ is rational, this is either known, or follows from existing results, and our contribution is a resolution of the irrational case. We deduce that for all $c \not \in \{0, 1, 2\}$, the cardinality of the sumset $S + S$ asymptotically attains its natural upper bound $N(N + 1)/2$, as $N \to \infty$. We show that there are infinitely many, effectively computable numbers $c$ such that the set $\{p^c : \textrm{$p$ prime}\}$ is additively dissociated (actually linearly independent over $\mathbb{Q}$), and we provide an effective procedure to compute the digits of such $c$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04081 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive relations in irrational powers Harrison, Joseph Number Theory Combinatorics We investigate the additive theory of the set $S = \{1^c, 2^c, \dots, N^c\}$ when $c$ is a real number. In the language of additive combinatorics, we determine the asymptotic behaviour of the additive energy of $S$. When $c$ is rational, this is either known, or follows from existing results, and our contribution is a resolution of the irrational case. We deduce that for all $c \not \in \{0, 1, 2\}$, the cardinality of the sumset $S + S$ asymptotically attains its natural upper bound $N(N + 1)/2$, as $N \to \infty$. We show that there are infinitely many, effectively computable numbers $c$ such that the set $\{p^c : \textrm{$p$ prime}\}$ is additively dissociated (actually linearly independent over $\mathbb{Q}$), and we provide an effective procedure to compute the digits of such $c$. |
| title | Additive relations in irrational powers |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2512.04081 |