Conjecture 4.21

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\Gamma\subset(\overline{\mathbb Q}\cap\mathbb R_{>0})^n$ be a finitely generated multiplicative subgroup dense in $(\mathbb R_{>0})^n$. Fix generators $\gamma_j=(\gamma_{1j},\ldots,\gamma_{nj})$, $1\le j\le m$. For $s\in\mathbb Z^m$ put $\gamma_i(s)=\prod_{j=1}^m\gamma_{ij}^{s_j}$ and $|s|_\infty=\max_j|s_j|$. Fix a compact neighborhood $\mathcal K$ of $(1,\ldots,1)$ in $(\mathbb R_{>0})^n$.

Statement recorded in the supplied source.

For every $\varepsilon>0$, does there exist $S_0>0$, depending only on $\varepsilon$, the fixed generators and $\mathcal K$, such that, for every real $S\ge S_0$ and every $\zeta=(\zeta_1,\ldots,\zeta_n)\in\mathcal K$, there is $s\in\mathbb Z^m$ with \[ |s|_\infty\le S,\qquad \max_{1\le i\le n}|\gamma_i(s)-\zeta_i|\le S^{-1-1/n+\varepsilon}? \] [S2]

Short English statement

Does the dense algebraic multiplicative subgroup approximate every target in a fixed compact neighborhood with bounded generator exponents and the stated uniform error rate?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700063 (AMR-086-0063). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 4.21 and immediately preceding definitions, p.44. https://arxiv.org/abs/math/0312440v2.