By Dzmitry Badziahin, Alexander Gorodnik, Norbert Peyerimhoff

ISBN-10: 1107552370

ISBN-13: 9781107552371

Written by way of prime specialists, this ebook explores a number of instructions of present learn on the interface among dynamics and analytic quantity concept. themes comprise Diophantine approximation, exponential sums, Ramsey idea, ergodic conception and homogeneous dynamics. The origins of this fabric lie within the 'Dynamics and Analytic quantity concept' Easter tuition held at Durham collage in 2014. Key recommendations, state of the art effects, and glossy ideas that play an important position in modern learn are offered in a fashion obtainable to younger researchers, together with PhD scholars. This booklet can be helpful for confirmed mathematicians. The parts mentioned contain ubiquitous structures and Cantor-type units in Diophantine approximation, flows on nilmanifolds and their connections with exponential sums, a number of recurrence and Ramsey conception, counting and equidistribution difficulties in homogeneous dynamics, and functions of skinny teams in quantity concept. either dynamical and 'classical' ways in the direction of quantity theoretical difficulties also are supplied.

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Extra resources for Dynamics and Analytic Number Theory

Example text

23) t=1 Then, for any x = (x1 , . . , xn ) ∈ Rn and N ∈ N, there exists q ∈ Z such that max{ q x 1 1/i 1 , . . , q xn 1/i n } < N −1 and 1 ≤ q ≤ N . 2 The symmetric case corresponding to i 1 = . . = i n = 1/n is the more familiar form of the theorem. In this symmetric case, when N is an nth power, the one-dimensional proof using the Pigeonhole principle can easily be adapted to prove the associated statement (exercise). 1 below. , i 1 qN q N in respectively. Now the shape of the rectangles are clearly governed by (i 1 , .

9 Let n = 2 and note that if (x, y) ∈ Bad(i, j ) for some pair (i, j), then it would imply that lim inf q q x q→∞ qy = 0. Hence ∩i+ j =1 Bad(i, j) = ∅ would imply that Littlewood’s conjecture is true. 4. 10 Geometrically speaking, Bad(i 1 , . . , i n ) consists of points x ∈ Rn that avoid all rectangles of size ci1 q −(1+i1 ) × . . × cin q −(1+in ) centred at rational points ( p1 /q, . . , pn /q) with c = c(x) sufficiently small. Note that in the symmetric case i 1 = . . = i n = 1/n, the rectangles are squares (or essentially balls), and this makes a profound difference when investigating the ‘size’ of Bad(i 1 , .

The problem in the above example is that the sets E q overlap ‘too much’ – in fact, they are nested. The upshot is that in order to have μ(E ∞ ) > 0, we not only need the sum of the measures to diverge but also that the sets E q (q ∈ N) are in some sense independent. e. if μ(E s ∩ E t ) = μ(E s )μ(E t ) ∀s = t, then we would have μ(E ∞ ) = μ( ). However, we very rarely have this strong form of independence in our applications. What is much more useful to us is the following statement, whose proof can be found in [58, 90].

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Dynamics and Analytic Number Theory by Dzmitry Badziahin, Alexander Gorodnik, Norbert Peyerimhoff

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