Asymptotic formulas for sums of elements from a multiplicative group
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913171934019584 |
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| author | Evertse, Jan-Hendrik Győry, Kálmán Hajdu, Lajos Luca, Florian Remete, László |
| author_facet | Evertse, Jan-Hendrik Győry, Kálmán Hajdu, Lajos Luca, Florian Remete, László |
| contents | Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $Γ$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(α)$ denote the absolute exponential height of an algebraic number $α$. Fix non-zero elements $a_1,\ldots , a_k\in K$. We give asymptotic formulas for the number of $\mathbf{x}=(x_1,\ldots , x_k)\inΓ$ with $H(a_1x_1+\cdots +a_kx_k)\leq X$ as $X\to\infty$ such that no non-empty subsum of $a_1x_1+\cdots +a_kx_k$ vanishes. By the same method of proof, we obtain an asymptotic formula as $X\to\infty$ for the number of non-negative integers $n$ with $H(u_n)\leq X$, where $\{ u_n\}$ is a linear recurrence sequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28973 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic formulas for sums of elements from a multiplicative group Evertse, Jan-Hendrik Győry, Kálmán Hajdu, Lajos Luca, Florian Remete, László Number Theory 11D09, 11D61, 11B37 Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $Γ$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(α)$ denote the absolute exponential height of an algebraic number $α$. Fix non-zero elements $a_1,\ldots , a_k\in K$. We give asymptotic formulas for the number of $\mathbf{x}=(x_1,\ldots , x_k)\inΓ$ with $H(a_1x_1+\cdots +a_kx_k)\leq X$ as $X\to\infty$ such that no non-empty subsum of $a_1x_1+\cdots +a_kx_k$ vanishes. By the same method of proof, we obtain an asymptotic formula as $X\to\infty$ for the number of non-negative integers $n$ with $H(u_n)\leq X$, where $\{ u_n\}$ is a linear recurrence sequence. |
| title | Asymptotic formulas for sums of elements from a multiplicative group |
| topic | Number Theory 11D09, 11D61, 11B37 |
| url | https://arxiv.org/abs/2605.28973 |