Conjecture 4.20

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.

Write $\mathbb R_{>0}$ for the multiplicative group of positive real numbers. Fix positive integers $m,n,k$ and integers $a_{ij\kappa}$, where $1\le i\le n$, $1\le j\le m$, $1\le\kappa\le k$. For $x=(x_1,\ldots,x_k)\in(\mathbb R_{>0})^k$ define \[ \Gamma(x)=\left\{\left(\prod_{j=1}^m\prod_{\kappa=1}^k x_\kappa^{a_{ij\kappa}s_j}\right)_{i=1}^n:(s_1,\ldots,s_m)\in\mathbb Z^m\right\}. \] Here the product uses the $\kappa$th coordinate $x_\kappa$, in agreement with the parametrized-torus construction surrounding the displayed formula. Density uses the usual real topology. Positive numbers $\gamma_1,\ldots,\gamma_k$ are multiplicatively independent if $\prod_\kappa\gamma_\kappa^{b_\kappa}=1$ for integers $b_\kappa$ only when all $b_\kappa=0$.

Statement recorded in the supplied source.

If there exists $x\in(\mathbb R_{>0})^k$ for which $\Gamma(x)$ is dense in $(\mathbb R_{>0})^n$, is $\Gamma(\gamma)$ dense in $(\mathbb R_{>0})^n$ for every multiplicatively independent tuple of positive algebraic numbers $\gamma=(\gamma_1,\ldots,\gamma_k)$? [S2]

Short English statement

If this integer-exponent construction yields a dense subgroup for some positive real parameters, must it also be dense for every multiplicatively independent positive algebraic parameter tuple?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700062 (AMR-086-0062). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 4.20 and preceding torus formulation, p.43. https://arxiv.org/abs/math/0312440v2.