In the paper we find new inequalities involving the intersections $A\cap (A-x)$ of shifts of some subset $A$ from an abelian group. We apply the inequalities to obtain new upper bounds for the additive energy of multiplicative subgroups and convex sets and also a series another results on the connection of the additive energy and so--called higher moments of convolutions. Besides we prove new theorems on multiplicative subgroups concerning lower bounds for its doubling constants, sharp lower bound for the cardinality of sumset of a multiplicative subgroup and its subprogression and another results.

Some new inequalities in additive combinatorics

In the paper we develop the method of higher energies. New upper bounds for the additive energies of convex sets, sets A with small |AA| and |A(A+1)| are obtained. We prove new structural results, including higher sumsets, and develop the notion of dual popular difference sets.

/pdf/some-new-results-on-higher-energies-54h3ijhen9.pdf

Some new results on higher energies

Let $F$ be a field of characteristic $p>2$ and $A\subset F$ have sufficiently small cardinality in terms of $p$. We improve the state of the art of a variety of sum-product type inequalities. In particular, we prove that $$ |AA|^2|A+A|^3 \gg |A|^6,\qquad |A(A+A)|\gg |A|^{3/2}. $$ We also prove several two-variable extractor estimates: ${\displaystyle |A(A+1)| \gg|A|^{9/8},}$ $$ |A+A^2|\gg |A|^{11/10},\; |A+A^3|\gg |A|^{29/28}, \; |A+1/A|\gg |A|^{31/30}.$$ 
Besides, we address questions of cardinalities $|A+A|$ vs $|f(A)+f(A)|$, for a polynomial $f$, where we establish the inequalities $$ \max(|A+A|,\, |A^2+A^2|)\gg |A|^{8/7}, \;\; \max(|A-A|,\, |A^3+A^3|)\gg |A|^{17/16}. $$ Szemeredi-Trotter type implications of the arithmetic estimates in question are that a Cartesian product point set $P=A\times B$ in $F^2$, of $n$ elements, with $|B|\leq |A|< p^{2/3}$ makes $O(n^{3/4}m^{2/3} + m + n)$ incidences with any set of $m$ lines. In particular, when $|A|=|B|$, there are $\ll n^{9/4}$ collinear triples of points in $P$, $\gg n^{3/2}$ distinct lines between pairs of its points, in $\gg n^{3/4}$ distinct directions. 
Besides, $P=A\times A$ determines $\gg n^{9/16}$ distinct pair-wise distances. 
These estimates are obtained on the basis of a new plane geometry interpretation of the incidence theorem between points and planes in three dimensions, which we call collisions of images.

/pdf/growth-estimates-in-positive-characteristic-via-collisions-6uc5dnj6hf.pdf

Growth Estimates in Positive Characteristic via Collisions

On exponential sums over multiplicative subgroups of medium size

In the paper we prove that any sumset or difference set has large E_3 energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms. In particular, we give criteria for a set to be a 1) set of the form H+L, where H+H is small and L has "random structure", 2) set equals a disjoint union of sets H_j, each H_j has small doubling, 3) set having large subset A' with 2A' is equal to a set with small doubling and |A'+A'| \approx |A|^4 / \E(A).

Energies and structure of additive sets

It is proved that for an arbitrary subgroup and any distinct nonzero elements we have under the condition that , where ,  are some sequences of positive numbers such that as . Furthermore, it is shown that the inequality holds for any subgroup  such that . Bibliography: 25 titles.

On additive shifts of multiplicative subgroups

/pdf/ob-additivnykh-sdvigakh-multiplikativnykh-podgrupp-on-2sdb1foaw8.pdf

Об аддитивных сдвигах мультипликативных подгрупп@@@On additive shifts of multiplicative subgroups

Several inverse problems of the analytic theory of differential equations are considered: an estimate of the number of extra singular points occurring in the construction of a Fuchsian equation for an arbitrary (for instance, reducible) monodromy representation is found; an estimate of the Poincare rank of the unique non-Fuchsian singularity of the regular linear system constructed for an arbitrary monodromy representation is obtained; the problem of the meromorphic reduction to polynomial form of a linear system in the neighbourhood of an irregular singularity is investigated (which is related to the reduction of a linear system to a Birkhoff standard form).

/pdf/additional-parameters-in-inverse-monodromy-problems-3hg74xwmyu.pdf

Additional parameters in inverse monodromy problems

/pdf/o-dopolnitelnykh-parametrakh-v-obratnykh-zadachakh-sbmy7pkolr.pdf

О дополнительных параметрах в обратных задачах монодромии@@@Additional parameters in inverse problems of monodromy

The main result of the paper is the formula that calculates the dispersion of the asymptotic Hopf invariant of a magnetic field. The paper contain examples, which describe magnetic fields in a conductive medium.

/pdf/dispersion-of-the-arnold-s-asymptotic-ergodic-hopf-invariant-34o0u49gqf.pdf

Dispersion of the Arnold’s Asymptotic Ergodic Hopf Invariant and a Formula for Its Calculation

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