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Ciprian Demeter

Publications and source records attributed to Ciprian Demeter.

At least 37 records · Page 2Linked to original sources
arXiv2020

Small cap decouplings

We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in $\mathbb{R}^3$. This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks. We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone. The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.

math.CA↗
arXiv2020

On the refined Strichartz estimates

We present a slightly simpler proof of the multilinear refined Strichartz estimate, and prove a slightly more general linear refined Strichartz estimate. Our arguments seek to clarify the connection between these estimates, refined decoupling and tube incidences.

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arXiv2019

Three applications of the Siegel mass formula

We present three applications of the Siegel mass formula. First we estimate the number of solutions of a quadratic system of equations. We also include estimates for the distribution of lattice points on caps of four dimensional spheres and for the number of non-congruent lattice tetrahedra.

math.NT↗
arXiv2018

On the restriction theorem for paraboloid in $\mathbb R^4$

We prove that recent breaking by Zahl of the $\frac32$ barrier in Wolff's estimate on the Kakeya maximal operator in $\mathbb R^4$ leads to improving the $\frac{14}{5}$ threshold for the restriction problem for the paraboloid in $\mathbb R^4$. One of the ingredients is a new trilinear estimate. The proofs are deliberately presented in a nontechnical and concise format, so as to make the arguments more readable and focus attention on the key tools.

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arXiv2016

A guide to Carleson's Theorem

This paper is meant to be a gentle introduction to Carleson's Theorem on pointwise convergence of Fourier series.

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arXiv2016

Linear independence of time frequency translates for special configurations

We prove that for any 4 points in the plane that belong to 2 parallel lines, there is no linear dependence between the associated time-frequency translates of any nontrivial Schwartz function. If mild Diophantine properties are satisfied, we also prove linear independence in the category of $L^2(\R)$ functions.

math.CA↗
arXiv2015

Decouplings for curves and hypersurfaces with nonzero Gaussian curvature

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from \cite{BD3}. As a consequence of this we obtain sharp (up to $ε$ losses) Strichartz estimates for the hyperbolic Schrödinger equation on the torus. Our second main result is an $l^2$ decoupling for non degenerate curves which has implications for Vinogradov's mean value theorem.

math.CA↗
arXiv2015

The proof of the $l^2$ Decoupling Conjecture

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which (up to $N^ε$ losses) implies the full range of expected $L^p_{x,t}$ Strichartz estimates for both classical and irrational tori. Another one is an improvement in the range for the discrete restriction theory for lattice points on the sphere. Various applications in Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory.

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