1
$\begingroup$

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a topic of Diophantine Approximation and there are lots of research relating Ergodic Theory and Diophantine Approximation. I even took a course and studied Ergodic Theory book by Walters and Ergodic Theory with a View towards Number Theory book by Einsiedlerhoping. Then I searched lots of papers but there is no sign of proving irrationality of specific constants by methods of Ergodic Theory. On the other hand, there are famous unsolved problems in Number Theory that are solved recently (fully or almost fully) by Ergodic Theory like the Littlewood's conjecture, Green-Tao theorem, Erdos Discrepancy Problem, etc. There are lots of papers also relating Ergodic Theory and Diophantine Approximation but nothing about proving irrationality of numbers.

My question is would you please cite a paper regarding proving irrationality of numbers through Ergodic Theory methods?

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ I don't really see a question here $\endgroup$
    – Asaf
    Commented Jun 14, 2021 at 20:27
  • $\begingroup$ @Asaf, thank you indeed for your reply. How can I edit the last line that I have asked my question? My question is actually was how can I find a paper that I cannot find if exists about the problem I asked and if there really is NO such papers why is so? I.e. is there a paper showing that ergodic theory is not applicable in that area of diophantine appr? $\endgroup$
    – user288447
    Commented Jun 14, 2021 at 20:33
  • 2
    $\begingroup$ Ask the question to the professor who guided you. Perhaps you had a misunderstanding, for instance. Rather than edit your last line, try shortening the earlier part or at least putting the less crucial parts of it after your question. Keep in mind that if X is proved by Y and Y is related to Z, it need not follow that X can be proved by Z. Ergodic theory is good at proving directly that almost all numbers have a property, for instance almost all numbers are normal. But nobody has proved a specific number is normal by ergodic theory (only by other methods, and only for boring numbers). $\endgroup$
    – KConrad
    Commented Jun 14, 2021 at 21:00
  • $\begingroup$ @KConrad, is it really hard to find a way to prove irrationality or normality of a specific number by ergodic theory? At least, how can I collect all literature about appearance of Euler's constant in ergodic theory? I wanted to write a proposal for my MSc thesis. $\endgroup$
    – user288447
    Commented Jun 14, 2021 at 21:19
  • 1
    $\begingroup$ Why do you think it should not be hard when you can't find it done anywhere? Read mathoverflow.net/questions/129364/…. As Henry Cohn wrote in a comment there, "all irrationality questions are hard by default, and it's $e$ and $\pi$ that are special in being unusually tractable." No a priori interesting number has ever been proved normal by any method at all. See arxiv.org/abs/1303.1856 and search for "ergodic"; sure looks like slim pickings (you can google that phrase if you're not familiar with it). $\endgroup$
    – KConrad
    Commented Jun 14, 2021 at 21:34

0

Reset to default

You must log in to answer this question.