Question 5.2 — Bugeaud

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.

Fix $n\ge2$. Define $\mathrm{ZH}_n\subset\mathbb R$ to consist of those $\xi$ for which constants $c_1(\xi),c_2(\xi)>0$ exist satisfying both: every algebraic $\alpha\in\mathbb C$ of degree at most $n$ obeys $|\xi-\alpha|\ge c_2(\xi)H(\alpha)^{-n-1}$, and infinitely many distinct algebraic $\alpha$ of degree at most $n$ obey $|\xi-\alpha|\le c_1(\xi)H(\alpha)^{-n-1}$. The absolute value is complex modulus.

Statement recorded in the supplied source.

For every $n\ge2$, is \[ \{\xi\in\mathbb R:[\mathbb Q(\xi):\mathbb Q]=n+1\}\ \subsetneq\ \mathrm{ZH}_n? \] [S2]

Short English statement

For each degree bound, is the set having both uniform lower bounds and infinitely many matching upper approximations strictly larger than the real algebraic numbers of the next degree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700065 (AMR-086-0065). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Question 5.2, p.46; usual-height convention, p.35. https://arxiv.org/abs/math/0312440v2.