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Originally posted on mathoverflow but didn't get an answer.

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or ζ(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 evaulation of irrationality of specific numbers through Ergodic Theory methods?

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    That is still an open question, that is probably why no one provided an answer. There is a nice article by M. R. Murty (his brother also works in number theory) that you may want to check. It relates to some special values of the Gamma function where you may get an idea of the techniques that are used (from analytical to algebraic) – Mittens Jun 17 '21 at 17:31
  • @OliverDiaz, I know that's an open problem. What happened is that I spent over two years studying Euler-Mascheroni and learned close and remote methods even those applied for constants of totally different arithmetical nature. Then out of blue a professor suggested me that Ergodic has solve many Diophantine approx problems maybe I should try that way but this new subject lack the constant entirely, i.e. not even the very first attempts! –  Jun 17 '21 at 18:32
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    aas a problem aside for a beginner it will give you a lot of stuff to try and learn. But If you want that to be your goal for your Master or even Ph. D. thesis just tread carefully. Big names are likely working if not on this problem per se, in there problems that will have the irrationality (or rationality) of $\gamma$ as a bio-product. – Mittens Jun 17 '21 at 18:50
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    Did you look at https://arxiv.org/pdf/1303.1856.pdf, which was mentioned in a comment to your MO post, especially Section 3.16? Theorem 3.16.3 is about an infinite family of numbers $\gamma(h,k)$ where $\gamma = 4\gamma(2,4)$. Murty and Saradha proved all but at most one of the numbers $\gamma(h,k)$ is transcendental (surely they are all transcendental even if that has not yet been proved). The proof is not based on ergodic theory. Be careful about developing an obsession to settle a famous problem in your MSc thesis by a method that has not been used before in a similar way. – KCd Jun 19 '21 at 18:26
  • In order to prove a desired theorem it may be better to prove a stronger theorem. Examples of this are in https://math.stackexchange.com/questions/1007256/examples-where-it-is-easier-to-prove-more-than-less. Irrationality is weaker than transcendence, and maybe irrationality of $\gamma$ will first be proved by showing it is transcendental. There are examples where certain numbers (defined in a complicated way) are known to be nonzero by proving they are transcendental: see https://mathoverflow.net/questions/291723/regulator-of-an-elliptic-curve-rational-irrational-transcendental. – KCd Jun 19 '21 at 18:36
  • @KCd, I've spent over 4 months on that article and also two of the names in the literature (the couple) Kh H & T H Pilehrood were my BSc thesis advisor (and they don't work on Ergodic). I am looking for a collection of even most elementary attempts connecting Euler's constant and Ergodic but beyond a simple appearance of it. –  Jun 19 '21 at 18:41
  • @KCd, many constants that I know are proved to be transcendental first were proved to be irrational. See Making Transcendental Transparent by Burger for some of them, for example. But yes generalized Euler's constant has been explored much more than the gamma –  Jun 19 '21 at 18:43
  • I should have included Theorem 3.16.4 in my previous comment as providing another list of numbers containing $\gamma$ where all but at most one number in the list is provably transcendental. I agree that the famous classical transcendental numbers like $e$ and $\pi$ were proved to be irrational before they were proved to be transcendental, but I am not persuaded that it's realistic to expect this to wind up working out for $\gamma$. For the numbers $\gamma(h,k)$, I am unaware of any result on the irrationality of at least one that does not rely on proving transcendence of "many" of them. – KCd Jun 19 '21 at 19:00

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