Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_{\lambda} N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll \lambda^{1/4} (N\lambda)^{-\delta}$ (for some $\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.Comment: version 3: substantially revised versio

Estimating quadratic variation when quoted prices jump by a constant increment

Financial assets' quoted prices normally change through frequent revisions, or jumps. For markets where quotes are almost always revised by the minimum price tick, this paper proposes a new estimator of Quadratic Variation which is robust to microstructure effects. It compares the number of alternations, where quotes are revised back to their previous price, to the number of other jumps. Many markets exhibit a lack of autocorrelation in their quotes' alternation pattern. Under quite general 'no leverage' assumptions, whenever this is so the proposed statistic is consistent as the intensity of jumps increases without bound. After an empirical implementation, some useful corollaries of this are given.

A large covariance matrix estimator under intermediate spikiness regimes

The present paper concerns large covariance matrix estimation via composite minimization under the assumption of low rank plus sparse structure. In this approach, the low rank plus sparse decomposition of the covariance matrix is recovered by least squares minimization under nuclear norm plus $l_1$ norm penalization. This paper proposes a new estimator of that family based on an additional least-squares re-optimization step aimed at un-shrinking the eigenvalues of the low rank component estimated at the first step. We prove that such un-shrinkage causes the final estimate to approach the target as closely as possible in Frobenius norm while recovering exactly the underlying low rank and sparsity pattern. Consistency is guaranteed when $n$ is at least $O(p^{\frac{3}{2}\delta})$, provided that the maximum number of non-zeros per row in the sparse component is $O(p^{\delta})$ with $\delta \leq \frac{1}{2}$. Consistent recovery is ensured if the latent eigenvalues scale to $p^{\alpha}$, $\alpha \in[0,1]$, while rank consistency is ensured if $\delta \leq \alpha$. The resulting estimator is called UNALCE (UNshrunk ALgebraic Covariance Estimator) and is shown to outperform state of the art estimators, especially for what concerns fitting properties and sparsity pattern detection. The effectiveness of UNALCE is highlighted on a real example regarding ECB banking supervisory data

Energy Loss of a Heavy Quark Produced in a Finite Size Medium

We study the medium-induced energy loss $-\Delta E_0(L_p)$ suffered by a heavy quark produced at initial time in a quark-gluon plasma, and escaping the plasma after travelling the distance $L_p$. The heavy quark is treated classically, and within the same framework $-\Delta E_0(L_p)$ consistently includes: the loss from standard collisional processes, initial bremsstrahlung due to the sudden acceleration of the quark, and transition radiation. The radiative loss {\it induced by rescatterings} $-\Delta E_{rad}(L_p)$ is not included in our study. For a ultrarelativistic heavy quark with momentum p \gsim 10 {\rm GeV}, and for a finite plasma with L_p \lsim 5 {\rm fm}, the loss $-\Delta E_0(L_p)$ is strongly suppressed compared to the stationary collisional contribution $-\Delta E_{coll}(L_p) \propto L_p$. Our results support that $-\Delta E_{rad}$ is the dominant contribution to the heavy quark energy loss (at least for L_p \lsim 5 {\rm fm}), as indeed assumed in most of jet-quenching analyses. However they might raise some question concerning the RHIC data on large $p_{\perp}$ electron spectra.Comment: 18 pages, 3 figures. New version clarified and simplified. A critical discussion added in section 2, and previous sections 3 and 4 have been merged together. Main results are unchange

On some mean value results for the zeta-function in short intervals

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x) := -\Delta(x) + 2\Delta(2x) - \frac{1}{2}\Delta(4x)$ and $\int_0^T E^*(t)\,dt = \frac{3}{4}\pi T + R(T)$, then we obtain a number of results involving the moments of $|\zeta(1/2+it)|$ in short intervals, by connecting them to the moments of $E^*(T)$ and $R(T)$ in short intervals. Upper bounds and asymptotic formulas for integrals of the form $$\int_T^{2T}\left(\int_{t-H}^{t+H}|\zeta(1/2+iu)|^2\,du\right)^k\,dt \qquad(k\in N, 1 \ll H \le T)$$ are also treated.Comment: 18 page