October 5, 2014
Similar papers 3
December 8, 2008
We extend the recent Solymosi's sum-product estimate for reals to the complex case.
November 5, 2021
The main results of this paper concern growth in sums of a $k$-convex function $f$. Firstly, we streamline the proof of a growth result for $f(A)$ where $A$ has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound for \[ |2^k f(A) - (2^k-1)f(A)|. \] We also generalise a recent result of Hanson, Roche-Newton and Senger, by proving that for any finite $A\subset \mathbb{R}$ \[ | 2^k f(sA-sA) - (2^k-1) f(sA-sA)| \g...
November 11, 2010
This note improves the best known exponent 1/12 in the prime field sum-product inequality (for small sets) to 1/11, modulo a logarithmic factor.
October 3, 2020
We study the $\delta$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $\alpha\in(1,\frac{3}{2}]$, we prove that for any $\sim|A|^{-1}$-separated set $A\subset[1,2]$ and $\delta=|A|^{-\alpha}$, we have: $\mathcal{N}(A+A, \delta)\cdot \mathcal{N}(AA, \delta) \gtrsim_{\epsilon}|A|\delta^{-1+\epsilon}$.
May 13, 2014
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 ...
May 26, 2018
We obtain an upper bound for the multiplicative energy of the spectrum of an arbitrary set from $\mathbb{F}_p$, which is the best possible up to the results on exponential sums over subgroups.
March 19, 2015
We improve a result of Solymosi on sum-products in R, namely, we prove that max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets A from R with |AA| \le |A|^{4/3}.
August 3, 2020
We obtain a series of lower bounds for the product set of combinatorial cubes, as well as some non--trivial upper estimates for the multiplicative energy of such sets.
June 27, 2024
Recent advances have linked various statements involving sumsets and cardinalities with corresponding statements involving sums of random variables and entropies. In this vein, this paper shows that the quantity $2{\bf H}\{X, Y\} - {\bf H}\{X+Y\}$ is a natural entropic analogue of the additive energy $E(A,B)$ between two sets. We develop some basic theory surrounding this quantity, and demonstrate its role in the proof of Tao's entropy variant of the Balog--Szemer\'edi--Gower...
February 19, 2015
In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log |A|}$. The latter inequality had in fact already been established in an earlier work of the author and Rudnev (arXiv:1203.6237), which built upon the recent developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s dist...