Conjecture 5.3

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 number $\alpha$, choose its primitive irreducible polynomial $f_\alpha\in\mathbb Z[T]$ with positive leading coefficient $a$ and roots $\alpha^{(1)},\ldots,\alpha^{(d)}$. Set \[ M(\alpha)=a\prod_{j=1}^d\max\{1,|\alpha^{(j)}|\},\qquad \mathrm h(\alpha)=\frac{\log M(\alpha)}d, \] and let $H(\alpha)$ be the maximum absolute value of the coefficients of $f_\alpha$. These are respectively Mahler measure, absolute logarithmic height, and usual polynomial height.

Almost all tuples here means outside a Lebesgue-null subset of $\mathbb R^n$, the real simultaneous-approximation setting of Section 5.1. Let $n\ge1$, $x=(x_1,\ldots,x_n)\in\mathbb R^n$, and write $F=\mathbb Q(\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n)$ for algebraic approximants.

Statement recorded in the supplied source.

For almost every $x\in\mathbb R^n$, do there exist constants $c,D_0>0$, allowed to depend on $n$ and $x$, such that, for every integer $D\ge D_0$, every real $\mu\ge D$, and every $\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n\in\overline{\mathbb Q}$ with \[ [F:\mathbb Q]\le D,\qquad [F:\mathbb Q]\max_{1\le i\le n}\{\mathrm h(\alpha_i),\mathrm h(\beta_i)\}\le\mu, \] one has \[ \max_{1\le i\le n}\{|x_i-\beta_i|,|e^{x_i}-\alpha_i|\}\ge\exp(-cD^{1/(2n)}\mu)? \] [S2]

Short English statement

For almost every real tuple, is simultaneous algebraic approximation of the tuple and its exponentials uniformly limited by the stated field-degree and height bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700066 (AMR-086-0066). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 5.3, p.47; real-tuple metric context, p.46; heights, p.35. https://arxiv.org/abs/math/0312440v2.