On two-point configurations in subsets of pseudo-random sets

Transcription

1 On two-point configurations in subsets of pseudo-random sets Elad Aigner-Horev 2 Department of Mathematics University of Hamburg Bundesstrasse 55 D Hamburg, Germany Hiệp Hàn 1,3 Instituto de Matemática e Estatística Universidade de São Paulo Rua do Matão 1010, São Paulo, Brazil Abstract We prove a transference type result for pseudo-random subsets of Z N that is analogous to the well-known Fürstenberg-Sárközy theorem. More precisely, let k 2 be an integer and let and be real numbers satisfying +( )/(2 k+1 3) > 1. Let Z N be a set with size at least N and linear bias at most N. Then, every A with relative density A / (log log N) 1 2 log log log log log N contains a pair of the form {x, x + d k } for some nonzero integer d. For instance, for squares, i.e., k = 2, and assuming the best possible pseudo-randomness = /2 our result applies as soon as > 10/11. Our approach uses techniques of Green as seen in [6] relying on a Fourier restriction type result also due to Green.

2 1 Introduction A classical result in additive combinatorics, proved independently by Sárközy [10] and Fürstenberg [4], states that subsets of the first N integers with positive density contains a pair which di er by a perfect k-th power, i.e., a pair {x, x + d k } for some d>0. As a quantitative version it is known due to [9,1] that this conclusion already holds for sets A of density (1) A /N (log N) c log log log log N for some c>0 In this note we consider the problem of extending (1) to hold for subsets of vanishing relative density of sparse pseudorandom subsets of Z N. This type of extensions are commonly calledtransference results in which an extremal problem known for dense objects is transferred or carried over to sparse objects taken from a well-behaved universe like a random or a pseudorandom set. We refer to [7,8,11,2,3,6] for further information. Qualitative transference of (1) tosparserandomsetswere first considered in [7]. Later on, Nguyen [8] proved that with high probability every relatively dense subset of random sets R [N] of size (N 1 1/k )containstheconfiguration{x, x + d k }, for some nonzero d. Up to a multiplicative constant this density attained for the random host is best possible. It is also interesting to note that Tao and Ziegler proved that the polynomial Szemerédi theorem also holds in the primes. Their proof relies heavily on pseudo-random properties of the primes, thus, can be seen in the scheme mentioned above. For a classical notion of pseudo-randomness defined by small nontrivial Fourier coe cients, however, nothing is known concerning extensions of (1) andourtheorem1.1 shall give the first nontrivial bound for this setting. Our main result. Before stating the result we require some notation. For the purposes of Fourier analysis we endow Z N 1 Supported by FAPESP (Proc. 2010/ ) and by CNPq (Proc /2012-4) and by NUMEC/USP. 2 elad.horev@math.uni-hamburg.de 3 hh@ime.usp.br

3 with the counting measure and, consequently, endow its dual group Z b N with the uniform measure. As a result, given a function f : Z N! C the Fourier transform of f is defined to be the function f b : Z b N! C given by f( ) b = P x2z f(x)e( x) where e(x) =e 2 ix/n. We write kfk u =sup b 06= 2Z b N f( ) b to denote magnitude of the second largest Fourier coe cient of f, and call kfk u the linear bias of f. Given f,g : Z N! C, the convolution of f and g is given by f g(x) = P y2z N f(y)g(x y). We identify a set with its characteristic function, i.e., if A Z N then A also denotes a 0, 1-function with A(x) =1ifand only if x 2 A. Finally, let Q k = {x k : x apple N 1/k,x is integer } denote the set of kth powers. Our main result reads as follows. Theorem 1.1 Let k 2 be an integer and let > > 0 be reals with +( )/(2 k+1 3) > 1. Then there is an n 0 such that for all N>n 0 the following holds. Let Z N satisfy N and k k u apple N, and let = (N) (log log N) 1 2 log log log log log N. Then, every subset A satisfying A contains the configuration {x, x + d} A where d 2 Q k. It is worth to note that the proof of Theorem 1.1 does not merely guarantee one desired configuration but many. Indeed, the number of such configurations found in the subset A is at least 2 Q k N 1, where = ( ) > 0if 6= 0. Up to ( ) this bound is clearly best possible. Moreover, let us emphasise that in Theorem 1.1 we can handle subsets of whose relative density is vanishing as N goes to infinity. This is due to the fact that we are transferring the quantitative version of the dense case of the Fürstenberg-Sárközy theorem, namely (1). Due to Parseval s equality the the parameter in Theorem 1.1 controlling the pseudo-randomness of satisfies /2. That is, one may think of = /2 asthough is as pseudo-random as possible. In this case, i.e., = /2, we 1 have that Theorem 1.1 is applicable as long as > 1. 2 k+2 5

4 2 Sketch of the proof of Theorem 1.1 Our approach follows that of Green [6]. Given = (N) we introduce functions,,and" depending on N so that " > 0. Given the set A and a set of frequencies ;6= S Z b N we know that there is a % = %(N) 2 ["/2, "] such that the Bohr set B = B(S, %) isaregular (see, e.g. [12], chapter 4.4). Define the function a : Z N! R given by a(x) = N (A B)(x) whichcanbeshowntohavecertainattributes B seen in characteristic functions of dense sets; that is kak`1 N and kak 1 apple 3, provided k ku 1 > 2("/(20 S 1/2 )) S. Applying(1) combinedwithavarnavidestypeargumentthen yields a lower bound on the of number of desired configurations in a (2) (a, Q k )= X a(x)a(x + d)q k (d) N Q k. x,d2z N Due to the convolution property and assuming that A contains less than N 1+1/k pairs of the form {x, x k },wehave (a, Q k ) < N X (3) A( ) b 2 Q c 2 k ( ) 2Z b N b B( ) 2 B N 1+1/k from which we then derive a contradiction to (2). To this end, we split the sum into two sums; one ranging over 2 S = Spec (A) ={ 2 Z b N : A( ) b } and another ranging over 62 S. It can be shown that b B( ) 2 B 2 1 apple 2" for all 2 S and S apple (2/ ) (2 2 )/( ) from which we conclude that for the first sum ranging over S is at most 2% S b A(0) 2 c Q k (0) apple 2 Q k /4. Handling the sum over b Z N \ S, however, is more complicated. In fact, its proof is a central part of the proof of Theorem 1.1 but due to space limitation we omit it here. Nevertheless, we mention that the main tools in the proof are Waring s theorem and a restriction type result due to Green [5] which is similar to that used in [6]. It is a close adaption of the restriction theorem due to Stein-Tomas [13]. We omit the details.

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