Download An Irregular Mind: Szemerédi is 70 by Noga Alon (auth.), Imre Bárány, József Solymosi, Gábor Sági PDF

By Noga Alon (auth.), Imre Bárány, József Solymosi, Gábor Sági (eds.)

Szemerédi's impression on modern arithmetic, specially in combinatorics, additive quantity concept, and theoretical machine technological know-how, is gigantic. This quantity is a party of Szemerédi's achievements and character, at the social gathering of his 70th birthday. It exemplifies his impressive imaginative and prescient and distinct frame of mind. a couple of colleagues and buddies, all most sensible experts of their fields, have contributed their most modern examine papers to this quantity. the subjects comprise extension and functions of the regularity lemma, the lifestyles of k-term mathematics progressions in quite a few subsets of the integers, extremal difficulties in hypergraphs idea, and random graphs, them all attractive, Szemerédi sort arithmetic. It additionally includes released money owed of the 1st , very unique and hugely winning Polymath tasks, one led by way of Tim Gowers and the opposite through Terry Tao.

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Example text

Let d ~ 1 be any integer, and let a = (aI , . . , a d) be an arbitrary d-dimensional vector with real coordinates. The sequence na (mod 1), n = 1,2 ,3, .. is uniformly distributed in the unit cube [0, 1)d, meaning . 1 11m N->oo N 1 = volume (R) l oo < T : ta E R (mod 1)} = volume (R) for any axes-parallel box R = h x ..

D) Finally, we can switch from fixed n to all n's: Let f (x) = f (Xl, ... , Xd) be a Lebesgue square-integrable function defined in the d-dimensional unit cube I d = [0, l)d; for notational simplicity we extend f over the entire d-dimensional space ~d periodically with period one. 26) ,1. I: f(y + jx) - , 1n j=o r f(u) dul ::; a(f) . J2ii . (log2 n + (l/ct + 1) 2+e lId y'n holds for all (x , y) E Wand all integers n ~ 1. 24). Remarks. 1(a) and (d) , we notice a loss of factor o (log2+e n), which is basically negligible compared to the dominant factor of y'n in the denominator.

I briefly elaborate on this. , discrepancy theory). A typical question (or rather a bunch of questions) in this theory goes as follows. What are the most uniformly distributed N-element point sets (say) in the d-dimensional unit cube [O,l]d with respect to natural "geomet ric" families of sets , such as (1) all axes-parallel boxes, (2) all rotated boxes, (3) all balls, (4) all convex subsets, and so on. These questions led to dozens of interesting theorems; I just briefly mention two. I start with a theorem ofW.

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