On the rational approximation to linear combinations of powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909957656412160 |
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| author | Kumar, Veekesh Prasad, Gorekh |
| author_facet | Kumar, Veekesh Prasad, Gorekh |
| contents | For a complex number $x$, $\Vert x\Vert:=\min\{|x-m|:m\in\mathbb{Z}\}$. Let $k\geq 1$ be an integer, and $K$ be a number field. Let $α_1,\ldots,α_k$ be algebraic numbers with $|α_i|\geq 1$ and let $d_i$ denotes the degree of $α_i$ for $1\leq i\leq k$. Set $d=d_1+\cdots+d_k$. In this article, we show that if the inequality $
0<\Vertλ_1 qα^n_1+\cdots+λ_k qα^n_k\Vert<\frac{θ^n}{q^{d+\varepsilon}} $
has infinitely many solutions in $(n, q,λ_1,\ldots,λ_k)\in \mathbb{N}^2\times (K^\times)^k$ with absolute logarithmic Weil height of $λ_i$ is small compared to $n$ and some $θ\in (0,1)$, then, in particular, the tuple $(λ_1 qα^n_1,\ldots, λ_k qα^n_k)$ is pseudo-Pisot, and at least one of $α_i$ is an algebraic integer. This result can be viewed as Roth's type theorem for linear combinations of powers of algebraic numbers over $\overline{\mathbb{Q}}$. The case $q=1$ was recently proved by Kulkarni, Mavraki, and Nguyen \cite{kul}, which is a generalization of Mahler's question proved in \cite{corv}. As a consequence of our result, we obtain the following generalization of this question: let $α>1$ be an algebraic number with $d=[\mathbb{Q}(α):\mathbb{Q}]$. For a given $\varepsilon>0$, if the inequality $$ 0<\Vertλqα^n\Vert<\frac{θ^n}{q^{d+\varepsilon}} $$ has infinitely many solutions in the tuples $(n,q,λ)\in \mathbb{N}^2\times K^\times$ with absolute logarithmic Weil height of $λ$ is small compared to $n$ and $θ\in (0,1)$, then some power of $α$ is a Pisot number. As an application of this result, we deduce the transcendence of certain infinite products of algebraic numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the rational approximation to linear combinations of powers Kumar, Veekesh Prasad, Gorekh Number Theory 11J68, 11J87(Primary), 11B37, 11R06(Secondary) For a complex number $x$, $\Vert x\Vert:=\min\{|x-m|:m\in\mathbb{Z}\}$. Let $k\geq 1$ be an integer, and $K$ be a number field. Let $α_1,\ldots,α_k$ be algebraic numbers with $|α_i|\geq 1$ and let $d_i$ denotes the degree of $α_i$ for $1\leq i\leq k$. Set $d=d_1+\cdots+d_k$. In this article, we show that if the inequality $ 0<\Vertλ_1 qα^n_1+\cdots+λ_k qα^n_k\Vert<\frac{θ^n}{q^{d+\varepsilon}} $ has infinitely many solutions in $(n, q,λ_1,\ldots,λ_k)\in \mathbb{N}^2\times (K^\times)^k$ with absolute logarithmic Weil height of $λ_i$ is small compared to $n$ and some $θ\in (0,1)$, then, in particular, the tuple $(λ_1 qα^n_1,\ldots, λ_k qα^n_k)$ is pseudo-Pisot, and at least one of $α_i$ is an algebraic integer. This result can be viewed as Roth's type theorem for linear combinations of powers of algebraic numbers over $\overline{\mathbb{Q}}$. The case $q=1$ was recently proved by Kulkarni, Mavraki, and Nguyen \cite{kul}, which is a generalization of Mahler's question proved in \cite{corv}. As a consequence of our result, we obtain the following generalization of this question: let $α>1$ be an algebraic number with $d=[\mathbb{Q}(α):\mathbb{Q}]$. For a given $\varepsilon>0$, if the inequality $$ 0<\Vertλqα^n\Vert<\frac{θ^n}{q^{d+\varepsilon}} $$ has infinitely many solutions in the tuples $(n,q,λ)\in \mathbb{N}^2\times K^\times$ with absolute logarithmic Weil height of $λ$ is small compared to $n$ and $θ\in (0,1)$, then some power of $α$ is a Pisot number. As an application of this result, we deduce the transcendence of certain infinite products of algebraic numbers. |
| title | On the rational approximation to linear combinations of powers |
| topic | Number Theory 11J68, 11J87(Primary), 11B37, 11R06(Secondary) |
| url | https://arxiv.org/abs/2512.11337 |