Searcharxiv⌕ Search

arXiv subjects

Ciprian Demeter

Publications and source records attributed to Ciprian Demeter.

At least 19 recordsLinked to original sources
arXiv2026

Near optimal three-fold additive energy bound for points on convex curves

Let $X\subset\mathbb{R}$ be finite and let $\gamma(t)=(t,f(t))$, where $f$ is strictly convex. We show that \[ J_3(\gamma(X)) =\#\{(x_1,\ldots,x_6)\in X^6:\sum_{i=1}^3\gamma(x_i)=\sum_{i=4}^6\gamma(x_i)\} \ll_{\epsilon}|X|^{3+\epsilon}. \] When specialized to the parabola, our result implies near-optimal estimates for the number of solutions to the diameter-free quadratic Vinogradov system. As a second application, we settle a conjecture from Krishnapur-Kurlberg-Wigman and Bombieri-Bourgain concerning lattice points on dilates of the unit circle. As a third application, we prove that $|A-A|\gg_\epsilon|A|^{5/3-\epsilon}$ and $|A+A|\gg_\epsilon|A|^{8/5-\epsilon}$ for any finite convex sequence $A\subset \mathbb{R}$.

math.CA↗
arXiv2026

Decoupling for AD-regular sets on the parabola

We improve the decoupling exponent for functions with spectrum inside AD-regular collections of arcs on the parabola. We achieve this by incorporating recent Szemer\'{e}di--Trotter-type estimates into the bootstrapping argument from \cite{BD}. As an application, our results complement, and in some cases improve earlier results \cite{chang2022decoupling} for arithmetic Cantor sets.

math.CA↗
arXiv2026

New discretised polynomial expander and incidence estimates

We present two applications of recent developments in incidence geometry. One is a $\delta$-discretised version of a particular `Elekes--R\'onyai' expander problem. The second application is an incidence estimate addressing the scenario when both tubes, squares and their shadings satisfy non-concentration assumptions.

math.CO↗
arXiv2025

Incidence estimates for quasi-product sets and applications

We use recent advances in the theory of Furstenberg sets to prove new incidence results of Szemer\'edi--Trotter strength for $\delta$-discretized structures with Cartesian product flavor. We use these results to make progress on a number of problems that include energy estimates and Fourier decay of fractal measures supported on curves, as well as various sum-product-like results governed by fractal dimension.

math.CA↗
arXiv2024

Maximal $\Lambda(p)$-subsets of manifolds

We construct maximal $\Lambda(p)$-subsets on a large class of curved manifolds, in an optimal range of Lebesgue exponents $p$. Our arguments combine restriction estimates and decoupling with old and new probabilistic estimates.

math.CA↗
arXiv2024

Szemer\'edi-Trotter bounds for tubes and applications

We prove sharp estimates for incidences involving planar tubes that satisfy packing conditions. We apply them to improve the estimates for the Fourier transform of fractal measures supported on planar curves.

math.CA↗
arXiv2024

On the $N$-set occupancy problem

We explore variants of the following open question: Split $[0,1]^2$ into $N^2$ squares with side length $1/N$. Is there a way to select $N$ such squares such that each line intersects only $O(1)$ of them?

math.CO↗
arXiv2024

Beyond canonical decoupling

We introduce two families of inequalities. Large ensemble decoupling is connected to the continuous restriction phenomenon. Tight decoupling is connected to the discrete Restriction conjecture for the sphere. Our investigation opens new grounds and answers some questions.

math.CA↗
arXiv2023

$L^2$ to $L^p$ bounds for spectral projectors on the Euclidean two-dimensional torus

We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include l2 decoupling, small cap decoupling, and estimates of exponential sums.

math.CA↗
arXiv2022

Fourier decay for curved Frostman measures

We investigate decoupling for Frostman measures supported on curves with nonzero curvature. We combine this tool with known lower bounds for Furstenberg sets to reprove Orponen's recent result for the parabola.

math.CA↗
arXiv2021

On restriction of exponential sums to hypersurfaces with zero curvature

We prove essentially sharp bounds for the $L^p$ restriction of weighted Gauss sums to monomial curves. Getting the $L^2$ upper bound combines the $TT^*$ method for matrices with the first and second derivative test for exponential sums. The matching lower bound follows via constructive interference on short blocks of integers, near the critical point of the phase function. This method is used to make the broader point that restriction to hypersurfaces is really sensitive to curvature. Our results here complement earlier results by the author and Langowski.

math.CA↗
arXiv2021

Restriction of exponential sums to hypersurfaces

We prove moment inequalities for exponential sums with respect to singular measures, whose Fourier decay matches those of curved hypersurfaces. Our emphasis will be on proving estimates that are sharp with respect to the scale parameter $N$, apart from $N^\epsilon$ losses. In a few instances, we manage to remove these losses.

math.CA↗
arXiv2021

On $L^{12}$ square root cancellation for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$

We prove sharp $L^{12}$ estimates for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$. We place Bourgain's progress on the Lindel"of hypothesis in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves.

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.

math.CA↗