Conjecture 2.14 — Mahler

Definitions and mathematical statement

For a real number $t$, write \[ \|t\|=\min_{j\in\mathbb Z}|t-j| \] for its distance to the nearest integer. This notation is not the ordinary absolute value. Throughout, $\log$ is the natural real logarithm. The input variable $a$ ranges over all ordinary integers $a\ge2$.

Does there exist one absolute real constant $c>0$ such that \[ \|\log a\|>a^{-c}\qquad\text{for every integer }a\ge2? \] Equivalently, does some fixed power of $a$ give a uniform lower bound on how closely the logarithm of an integer can approach any integer? The strict inequality and the coefficient one on the right are part of the stated normalization.

The constant must work simultaneously for every $a$; it cannot vary with the nearest integer or with a subsequence of inputs. No explicit value of $c$ is requested. Replacing "every" by "all sufficiently large" gives an equivalent existence question, because finitely many small values can be accommodated by increasing $c$, and $\log a$ is never an integer for $a\ge2$.

The nearest integer to $\log a$ has size comparable to $\log a$, so this concerns unusually close encounters between ordinary integers and the exponential sequence $e^j$. Qualitative nonvanishing of each individual distance is weaker than the required common power-law estimate. Bounds with an exponent depending on $a$ do not establish the conjecture.

Short English statement

Is the distance from the natural logarithm of any integer to the nearest integer bounded below by one fixed inverse power of that integer?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8700017 (AMR-086-0017), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Waldschmidt - Open Diophantine Problems (2004), Conjecture 2.14. Original source indexed by AMR-086-0017. https://arxiv.org/abs/math/0312440.