Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/dfc/2007/reports/shkredov_report-08.pdf Äàòà èçìåíåíèÿ: Sat Dec 27 16:04:24 2008 Äàòà èíäåêñèðîâàíèÿ: Sun Feb 13 21:58:29 2011 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: wmap

Pierre Deligne contest.
Shkredov Ilya Dmitrievich, report, 2008.

1. In 2008 I wrote several papers : On sumsets of dissociated sets, Online Journal of Analytic Combinatorics, 27 pages, submitted for publication, On a result of J. Bourgain (with S.V. Konyagin), Izvestiya of Russian Academy of Sciences, 33 pages, submitted for publication, On monochromatic solutions of some nonlinear equations in Z/pZ, Mat. Zametki, 6 pages, submitted for publication, On some two­dimensional configurations in dense sets, 97 pages, submitted for publication.

2. About our results. In the first paper we are studying some properties of subsets Q of sums of dissociated sets. (Recall that a set = {1 , . . . , || } is called dissociated if the equation |i| i i = 0, where i {0, ±1} has the only solution i = 0). The =1 exact upper bound for the number of solutions of the following equation q1 + · · · + qp = q
p+1

+ · · · + q2p ,

qi Q

(1)

in groups Fn is found. Using our approach, we easily prove a recent result of 2 J. Bourgain on sets of large exponential sums and obtain a tiny improvement of his theorem. Besides an inverse problem is considered in the article. Let Q be a set belonging a sumset of two dissociated sets such that equation (1) has many solutions. We prove that in the case the large proportion of Q is highly structured. In his excellent paper on sets without arithmetic progressions of length three J. Bourgain obtained a new upper bound for the density of these sets. One of the crucial moments of the paper was a new result on so­called sets of large exponential sums. In article "On a theorem of J. Bourgain" we show that his key result is sharp. We use methods of the first paper and some probabilistic constructions. Also we negatively answer on a natural question of Tom Sanders about Chang's theorem. Suppose we have an arbitrary finite coloring of natural numbers. Is it true that there are x, y Z such that x + y and xy have the same color? The last


(unsolved) question is a well­known problem of Ramsey theory. In the third paper we give a positive answer to the question in the group Z/pZ, where p is a prime number. A well­known result of Gowers asserts that any set A {1, . . . , N } of density at least 1/(log log N )ck , ck > 0 has an arithmetic progression of length k . It is easy to see that the last theorem implies that for an arbitrary finite set F Z there is an affine copy aF + b, where a, b Z which belongs to the set A. In the last article we consider several two­dimension generalizations of the result for some specific sets F . Let us formulate our main theorem. Let A {1, . . . , N }2 be a subset of two­dimensional grid of the cardinality |N |2 /(log log log log N )c , where c > 0 is an absolute constant. We prove that A contains a quadruple {(x, y ), (x, y + r), (x + r, y ), (x + 2r, y )} and also {(x, y ), (x + r, y + r), (x + r, y ), (x + 2r, y )} for some r = 0. Thus the set F here is {(0, 0), (0, 1), (1, 0), (2, 0)} or {(0, 0), (1, 1), (1, 0), (2, 0)}. Our result is a two­dimensional generalization of Gowers' theorem in the case k = 4. Also we obtain a similar statement for subsets A of the group (Z/pZ)n â (Z/pZ)n , where n is a positive integer, and p 5 is a prime number.

3. In the year I took part at the conferences : "Uniform distribution" (Marseille, France, January 21­25, 2008), "Building bridges" (Budapest, Hungary, August 5­9, 2008), "Discrete Rigidity Phenomena in Additive Combinatorics" (Berkeley, USA, November 3­7, 2008). I gave talks at Minskii number­theoretical Gorodoskoi seminar, Houston mathematical seminar, Additive Combinatorics and Ergodic Theory seminar at MSRI.

4. In our special course "Szemer´ edi's Theorem and Fourier analysis" we discussed some results of Additive Number Theory and Combinatorial Ergodic Theory. For example we proved classical ergodic H. Furstenberg's proof of Szemer´ theorem and revisited J. Bourgain's result on arithmetic progressions. I edi read a special course at Moscow Independent Institute about Green and Tao's result on arithmetic progressions in the primes and a course "Ordinary Differential Equations" for second year students at Moscow State University. My student Maxim Makarov studied new Elkin's construction of sets without arithmetic progressions of length three. He gave a talk at our joint seminar with N.G. Moshchevitin "Exponential sums and its applications".