We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the socalled Hardy-Littlewood majorant property. We derive this from a rather more general result which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a r...

In this paper, we study the geometric generation of the (0; m; 2)-nets. Based on some combinatorial methods, a generation property on (0; m; 2)-nets with an applications to a Sobol' sequence and the number of the (0; m; 2)-nets in equivalent sense are given. We also present some symmetry properties of the Sobol' sequence and the Roth-Zaremba sequence.

We improve the quantitative estimate for Roth’s theorem on threeterm arithmetic progressions, showing that if A ⊂ {1, . . . , N} contains no nontrivial three-term arithmetic progressions then |A| N(log logN)4/ logN . By the same method we also improve the bounds in the analogous problem over Fq [t] and for the problem of finding long arithmetic progressions in a sumset.

THEOREM IB (Roth [11]). For every natural number N^2, there exists a distribution 0* of N points in U such that I. L-l \DL[&;B(y)]\dy«L(\ogN) u Note that Theorems 1A and IB remain true in the trivial case L = 1. Suppose now that ^ is a distribution of N points in the unit torus U = [0, 1)*, where K 5= 2. Let A be a compact and convex body in U. For any real number A € (0, 1], any proper orthogo...

The spectral-null code S(n; k) of k-th order and length n is the union of n-tuples with 1 components, having k-th order spectral null at zero frequency. We determine the exact asymptotic in n behaviour of the size of such codes. In particular, we prove that for n satisfying some divisibility conditions, log 2 jS(n; k)j = n ? k 2 2 log 2 n + c k + o(1), where c k is a constant depending only on ...

We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and ⋆-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of 2× 2 block matrix representations of the equations a...

one of factors which distinguishes today’s organizations is having employees who are internally and psychologically empowered, so imcreasing it is an important program for all the pioneering organizations. in this research the effect of four-dimensional model of organizational justice - distributive justice, procedural justice, interpersonal justice and informational justice – on psychological ...