Roth’s theorem in many variables

We prove that if A ? {1, ..., N} has no nontrivial solution to the equation x ₁ + x ₂ + x ₃ + x ₄ + x ₅ = 5y, then \(|A| \ll Ne^{ - c(\log N)^{1/7} } \) , c > 0. In view of the well-known Behrend construction, this estimate is close to best possible.     

On the additive energy of the Heilbronn subgroup

A new upper bound for the additive energy of the Heilbronn subgroup is found. Several applications to the distribution of Fermat quotients are obtained.     

Intersections of shifts of multiplicative subgroups

Using Stepanov’s method, we obtain an upper bound for the cardinality of the intersection of additive shifts of several multiplicative subgroups of a finite field. The resulting inequality is applied to a question dealing with the additive decomposability of subgroups.     

On the pyatetskii-shapiro normality criterion for continued fractions

Analogues of the Pyatetskii-Shapiro normality criterion for continued fractions and for f-expandings with finite initial tiling are established, improving some results by Moshchevitin and Shkredov obtained in the 2002 paper “On Pyatetskii-Shapiro criterion of normality.”     

On a Generalization of Szemeredi's Theorem

Let N be a natural number and A [1, ..., N]² be a set of cardinalityat least is an absolute constant. We prove that A contains a triple {(k, m), (k+d, m), (k, m+d)},where d > 0. This theorem is a two-dimensional generalizationof Szemerédi's theorem on arithmetic progressions. 2000Mathematics Subject Classification 35J25, 37A15.     

On the Pyatetskii-Shapiro Criterion of Normality

In this paper, the Postnikov--Pyatetskii-Shapiro criterion of normality is given a more modern form. We obtain its unimprovable analogs for finite Markov chains, continued fractions, and generalized Bernoulli shifts.     

Some remarks on the Balog–Wooley decomposition theorem and quantities <Emphasis Type="Italic">D</Emphasis><Superscript>+</Superscript>, <Emphasis Type="Italic">D</Emphasis><Superscript>×</Superscript>

A finite element with new properties of approximation of higher derivatives is constructed, and a method for the construction of a finite element space in the planar case is proposed. The method is based on Yu.N. Subbotin’s earlier results as well as on the results obtained in this paper. The constructed piecewise polynomial function possesses the continuity property and new approximation properties.