Related papers: On a Restriction Problem of Hickman and Wright for…

Mockenhaupt and Tao (Duke 2004) proved a finite field analogue of the Stein--Tomas restriction theorem, establishing a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure $\mu$ on a vector space over a finite…

Let $(\Omega,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$ and $u_{\lambda}:= \phi_{\lambda} |_{\partial \Omega}$ the associated…

Let $\lambda$ the Barban--Vehov weights, defined in $(1)$. Let $X\ge z_1\ge100$ and $z_2=z_1^\tau$ for some $\tau>1$. We prove that \begin{equation*} \sum_{n\le X}\frac{1}{n}\Bigl(\sum_{\substack{d|n}}\lambda_d\Bigr)^2 \le f(\tau)\frac{\log…

Hardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $\mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by…

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer…

In this paper, we prove the restriction estimates for 2D surfaces S:= {(xi1, xi2, xi1^3 +/- xi2^3) : (xi1, xi2) in [0,1]^2} by reducing to Wang-Wu's result on the perturbed paraboloid and to the results on the perturbed hyperboloid obtained…

In this paper we study the restriction estimate for the flat disk over finite fields. Mockenhaupt and Tao initially studied this problem but their results were addressed only for dimensions $n=4,6$. We improve and extend their results to…

We study the natural resolution of the conjugated Haar multiplier $M_{w^{\frac{1}{2}}}T_{\sigma}M_{w^{-\frac{1}{2}}},$ where the multiplication operators $M_{w^{\pm\frac{1}{2}}}$ are decomposed into their canonical paraproduct…

For an $r$-uniform hypergraph $H$, let $\nu^{(m)}(H)$ denote the maximum size of a set~$M$ of edges in $H$ such that every two edges in $M$ intersect in less than $m$ vertices, and let $\tau^{(m)}(H)$ denote the minimum size of a collection…

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $\theta_p\in[0,\pi]$ to be the angle for which $a(p)=2p^{( k…

For any $0 < \alpha <1$, we construct Cantor sets on the parabola of Hausdorff dimension $\alpha$ such that they are Salem sets and each associated measure $\nu$ satisfies the estimate $\|{\widehat{f d\nu}}\|_{L^p(\mathbb{R}^2)} \leq C_p…

For a smooth $k$-dimensional submanifold $\Sigma$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(\Sigma)$ restriction bounds of Burq-G\'erard-Tzvetkov -- originally proved for individual Laplace--Beltrami…

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the…

Let $\mathcal{P}$ be a subset of primes and for each prime $p\in \mathcal{P}$, consider a subset $\mathcal{L}_p$ of $\mathbb{Z}/p\mathbb{Z}$. We provide restriction estimates with integers $\leq N$ sifted by…

Let $f$ be the Hecke eigenform for the modular group $SL_2(\mathbb{Z})$, and $L(s, \text{sym}^2 f)$ be the symmetric square $L$-function associated with $f$. For $\frac{1}{2}<\sigma<1$, define $m(\sigma)$ as the supremum of all numbers $m$…

Assuming the Riemann Hypothesis, we show that for $k>0$ $$ \frac{1}{T}\text{meas}\Big\{t\in [T,2T]:|\zeta(1/2+{\rm i} t)|>(\log T)^k\Big\}\leq C_k \frac{(\log T)^{-k^2}}{\sqrt{\log\log T}}, $$ where $C_k=\exp(e^{ck})$ for some absolute…

The Stein-Tomas restriction theorem on Euclidean space says one can meaningfully restrict $\hat{f}$ to the unit sphere of $\mathbb{R}^n$ provided $f \in L^p(\mathbb{R}^n)$ with $1 < p < 2$. This result can be rewritten in terms of the…

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for…

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise…

Horv\'ath and Kiss [Proc. Amer. Math. Soc., 2005] proved the upper bound estimate $\frac{\lambda _{n}}{\lambda _{m}}\leq \frac{n^{2}}{m^{2}}$ $ (n>m\geq 1) $ for Dirichlet eigenvalue ratios of the Schr\"odinger problem $-y''+q(x)y=\lambda…