Conjecture 4.12

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.

For an algebraic $\gamma$, define $h(\gamma)=\deg(\gamma)^{-1}\log M(\gamma)$ using its primitive minimal-polynomial Mahler measure. For an algebraic tuple $\gamma=(\gamma_1,\ldots,\gamma_m)$ put $d(\gamma)=[\mathbb Q(\gamma):\mathbb Q]$ and $\mu(\gamma)=d(\gamma)\max_i h(\gamma_i)$. Fix $\theta\in\mathbb C^m$ with $t=\operatorname{trdeg}_{\mathbb Q}\mathbb Q(\theta)\ge1$.

Statement recorded in the supplied source.

Do there exist $c_1,c_2>0$, depending on $\theta$, such that for every pair of real sequences $(D_\nu),(\mu_\nu)$ ($\nu\ge0$) satisfying $c_1\le D_\nu\le\mu_\nu$, $D_\nu\le D_{\nu+1}\le2D_\nu$, $\mu_\nu\le\mu_{\nu+1}\le2\mu_\nu$, and unbounded $\mu_\nu$, infinitely many $\nu$ admit an algebraic tuple $\gamma$ with \[d(\gamma)\le D_\nu,\qquad \mu(\gamma)\le\mu_\nu,\qquad \max_i|\theta_i-\gamma_i|\le\exp(-c_2D_\nu^{1/t}\mu_\nu)?\] [S2]

Short English statement

Can every tuple of positive transcendence degree be simultaneously approximated at the stated sharp rate along infinitely many indices of every controlled degree-and-height growth schedule?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700054 (AMR-086-0054). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 4.12 and preceding definition of mu. https://arxiv.org/abs/math/0312440.