# Additive energies on discrete cubes

@inproceedings{Pont2021AdditiveEO, title={Additive energies on discrete cubes}, author={Jaume de Dios Pont and Rachel Greenfeld and Paata Ivanisvili and Jos'e Madrid}, year={2021} }

We prove that for d ≥ 0 and k ≥ 2, for any subset A of a discrete cube {0, 1}, the k−higher energy of A (the number of 2k−tuples (a1, a2, . . . , a2k) in A with a1−a2 = a3−a4 = · · · = a2k−1−a2k) is at most |A|2 k+2), and log2(2 +2) is the best possible exponent. We also show that if d ≥ 0 and 2 ≤ k ≤ 10, for any subset A of a discrete cube {0, 1}, the k−additive energy of A (the number of 2k−tuples (a1, a2, . . . , a2k) in A with a1 +a2 + · · ·+ak = ak+1 +ak+2 + · · ·+a2k) is at most |A|2 ( 2k…

## Figures from this paper

## References

SHOWING 1-9 OF 9 REFERENCES

Energies and Structure of Additive Sets

- MathematicsElectron. J. Comb.
- 2014

It is proved that any sumset or difference set has large E_3 energy and a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms is given.

A Bound on Partitioning Clusters

- MathematicsElectron. J. Comb.
- 2017

The argument relies on establishing a one-dimensional convolution inequality that can be established by elementary calculus combined with some numerical verification that improves upon the trivial bound of $|X|^2$.

Convolution Estimates and Number of Disjoint Partitions

- MathematicsElectron. J. Comb.
- 2017

This paper extends the recent result of Kane-Tao, corresponding to the case $n=3$ where $c(3)\approx 1.725$ to an arbitrary finite number of disjoint $n-1$ partitions.

Decoupling for fractal subsets of the parabola

- Mathematics
- 2020

Abstract. We consider decoupling for a fractal subset of the parabola. We reduce studying lL decoupling for a fractal subset on the parabola tpt, tq : 0 ď t ď 1u to studying l2Lp{3 decoupling for the…

Additive combinatorics

- MathematicsCambridge studies in advanced mathematics
- 2007

The circle method is introduced, which is a nice application of Fourier analytic techniques to additive problems and its other applications: Vinogradov without GRH, partitions, Waring’s problem.

Inequalities for $$L^p$$-Norms that Sharpen the Triangle Inequality and Complement Hanner’s Inequality

- Mathematics
- 2018

In 2006 Carbery raised a question about an improvement on the na\"ive norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$…

Exploring the toolkit of Jean Bourgain

- Mathematics
- 2020

A selected number of tools are discussed here, and a case study is performed of how an argument in one of Bourgain's papers can be interpreted as a sequential application of several of these tools.