I am interested in the links between Ergodic Theory and Number Theory. Can anyone give some references for papers to read in this field? Any open problems? Or ideas where it may be applicable in NT?
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7$\begingroup$ ergodic theory with a view towards number theory is a nice book which might help you answer your questions $\endgroup$– mathworker21Commented Nov 9, 2023 at 14:15
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4$\begingroup$ I like Furstenberg's Recurrence in Ergodic Theory and Combinatorial Number Theory . Princeton University Press. $\endgroup$– F ZaldivarCommented Nov 9, 2023 at 14:29
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1$\begingroup$ see also math.stackexchange.com/q/1190757/87355 $\endgroup$– Carlo BeenakkerCommented Nov 9, 2023 at 14:32
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4 Answers
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For inspiration, you might enjoy reading The remarkable effectiveness of ergodic theory in number theory (2009).
The focus is on the use of the ergodic theory of homogenous flows to compute the distribution of $\{\sqrt n\}$ on the unit circle (Elkies-McMullen) and on the use of ergodic ideas to show that there are arbitrarily large arithmetic progressions of primes (Green-Tao). A third major line of research at the intersection of ergodic theory and number theory, rational approximations of irrational numbers, is discussed by Klaus Schmidt.
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The interplay between ergodic theory and Number Theory owes a lot to the Abel Prize winner Hillel Furstenberg. So, I must suggest his book, Recurrence in Ergodic Theory and Combinatorial Number Theory.
However, as a beginner, I think this note by Yufei Zhao is very useful.
Other researchers in field include, among others like Tao and Green, Sasha Fish, Björklund, Bulinski, Hildebrand and Balogh. You can check their papers out.
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Terence Tao has a nice blog post about looking at the collatz conjecture from the viewpoint of Ergodic Theory
There are links in the post to other papers which also might be of interest to you.
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The best example in my opinion is the work conducted mainly by Mariusz Lemańczyk, on problem related to the prime $k-$tuple conjecture. In the paper called $\mathscr{B}$-free sets and dynamics, prof. Lemańczyk together with Bartnicka, Kasjan and Kułaga-Przymus have proven that that, if the sequence $\mathscr{B} = \{b_1,b_2,\ldots\} \subset \mathbb{N}$ satisfies Erdös conditions, that is $\text{gcd}(b_i,b_j) = 1$ for $i \neq j$ and $\sum_{b \in \mathscr{B}}\frac{1}{b} < \infty$, then the set $$ \mathcal{F}_{\mathscr{B}} := \mathbb{Z} \setminus \bigcup_{b\ \in \mathscr{B}}b\mathbb{Z}$$ (called the set of $\mathscr{B}$-free numbers) has blocks of integers that appear infinitely often. For example, if we let $$\mathscr{B} = \{p^2 \colon p \in \mathbb{P}\},$$ then the set $\mathcal{F}_{\mathscr{B}}$ would be the set of square-free integers. And, because we have 2, 3, 5 being square-free numbers, then there are infinitely many integers $n$ for which $n+2,n+3,n+5$ are all square-free.