S. V. Konyagin

Systems of vectors in Euclidean space and an extremal problem for polynomials

n 112 _,x ll< (2) We use induction on n. For n = 1 our assertion is obvious. We will show that if it holds for all values less than n, then it holds for n. Suppose ~ = {xl,...,xn} is an arbitrary L-system. For i = i, . . ., n we put A i = { ] ~ { t . . . . . n}: ] • i , ( x ~ , x j ) ~ 0 } . (3 ) We c o n s i d e r two c a s e s . The f i r s t c a s e : f o r e a c h i = 1 , . . . , n t h e c a r d i n a l i t y o f A i i s a t m o s t n 2 / 3 . T h e n n 2 n I ] ~ I Xi[t = 2 i=1 (Zt, Xji ) = ~ = i ( xi Xi) + ~i~j( xi Xj)= = n J r ~ . # j , (xi, xp§ (xt, xr ---n + ~ = 1 2 J E A i (xi Xj) = n n =n + 21=1(xi, ~,j~AiXj)

On the Freiman theorem in finite fields

For brevity, we write F2 instead of F {1,...,n} 2 . If J ⊂ I, then there is a natural embedding of F2 in F2 (defined by setting xi = 0 for i ∈ I J). In particular, form ≤ n, we consider F2 as a subset of F2 . LetK ≥ 1. Introduce F (K) as the minimal constant such that, for any finite set I and any nonempty A ⊂ F2 with a doubling constant σ(A) ≤ K, there is an affine subspace V ⊂ F2 containing A and of cardinality |V | ≤ F (K)|A|. The assertion claiming that F (K) is finite for anyK ≥ 1 is referred to as the Freiman theorem in F2. Ruzsa [2] proved that F (K) ≤ 2K4 for any K ≥ 1. The upper bounds for F (K) were refined in a series of papers (see [3]–[5]). Green and Tao [6] recently found the asymptotics for the logarithm of F (K) asK → ∞ by establishing the relation F (K) = 2 √ K , K ≥ 2. (1) Moreover, as was noted in [6], F (K) ≥ 22K−O(logK), K ≥ 2. (2) Here and below, log stands for the logarithm to base 2. In the present note, we refine the remainder in (1).

Extremum Problems for Functions with Small Support

In this paper we consider an extremum problem for even periodic functions having small intervals as their supports and subject to restrictions on the sum of the absolute values of their Fourier coefficients.

Integral points on strictly convex closed curves

A negative answer is given to Swinnerton-Dyers question: Is it true that for any ε > 0 there exists a positive integer n such that for any planar closed strictly convex n-times differentiable curve Γ, when it is blown up a sufficiently large number ν of times, the number of integral points on the resultant curve will be less than νɛ. An example has been constructed when this number for an infinite number ν is not less than ν1/2, while Γ is infinitely differentiable.

Comparison of the >L1-Norms of Total and Truncated Exponential Sums

The paper is concerned with a conjecture stated by S. V. Bochkarev in the seventies. He assumed that there exists a “stability” for the L1-norm of trigonometric polynomials when adding new harmonics. In particular, the validity of this conjecture implies the well-known Littlewood inequality. The disproof of a statement close to Bochkarevs conjecture is given. For this, the following method is used: the L1-norm of a sum of one-dimensional harmonics is replaced by the Lebesgue constant of a polyhedron of sufficiently high dimension.

Recurrence of the integral of an odd conditionally periodic function

We prove that the integral of a sufficiently smooth odd conditionally periodic function with zero mean and incommensurable frequencies recurs. Furthermore, we obtain the lower and upper bounds for smoothness guaranteeing the recurrence of the integral.

Rate of divergence of some integrals

AbstractLower bounds for the absolute values of the functionsn

Divergence in measure of multiple Fourier series

M(x) = sumnolimits_{n leqslant x} {mu (n)}