10 - Conjectures and open questions

Summary

We gather open problems encountered in the preceding chapters with several new ones. Instead of formulating them in terms of questions, we merely prefer to propose statements whose validity remains open. In many cases there is no evidence for, or against, the assertion claimed. We do not recall the partial results obtained towards these problems, since they can be (hopefully) easily found in the present book.

A large list of open problems in the general theory of distribution modulo one has been compiled by Strauch and Nair [677]; see also the monograph [678].

The thematic ordering of the problems essentially follows Chapters 1 to 9.

The first problem was posed by Hardy [333] in 1919.

Problem 10.1. Are there a transcendental number α and a positive real number ξ such that ∥ξα_n∥ tends to 0 as n tends to infinity?

Very little is known on the sequence of fractional parts of e.

Problem 10.2. To prove that ∥eⁿ∥ does not tend to 0 as n tends to infinity?

The next problem is usually attributed to Mahler although it does not seem to have been stated explicitly in his papers.

Problem 10.3. To prove that there exists a positive real number c such that ∥eⁿ>∥ > e^−cn, for every n ≥ 1.

Waldschmidt [723] conjectured that a stronger result holds, namely that there exists a positive real number c such that ∥eⁿ∥ > n^−c for every n ≥ 1. This is supported by metrical results [391].