Related papers: Large Deviations and the Lukic Conjecture
Following ideas introduced by Beardon-Minda and by Baribeau-Rivard-Wegert in the context of the Schwarz-Pick lemma, we use the iterated hyperbolic difference quotients to prove a multipoint Julia lemma. As applications, we give a sharp…
We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet…
We prove a recent conjecture by Ulas on reducible polynomial substitutions.
The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm of a product of random matrices. This can be extended under various assumptions, covering nonnegative as well as invertible matrices, to a law of…
Ordinary differential equations of the first order on the torus have been investigated in detail by H. Poincar\'e and A. Denjoy. The long-standing problem of generalising these results for the equations of the order $k>1$ (or for the…
We prove a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles that includes biorthogonal Laguerre, Jacobi and Hermite ensembles, the matrix model of Lueck, Sommers…
Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that \vspace{1mm} \noindent {\bf Conjecture (Chowla).} {\em \begin{equation} \label{a.1} \sum_{n\le x} \lambda (f(n)) =o(x)…
We study the shifted convolution sums associated to completely multiplicative functions taking values in $\{\pm 1\}$ and give combinatorical proofs of two recent results in the direction of Chowla's conjecture. We also determine the…
In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these…
We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences $\sum_{i=1}^k a_i x_i \equiv b \Mod{n}$. In particular, we obtain explicit expressions…
Let $(m_1, m_2)$ be a pair of positive integers. Denote by $\mathbb{P}^1$ the complex projective line, and by $\mathbb{P}^1_{m_1,m_2}$ the orbifold complex projective line obtained from $\mathbb{P}^1$ by adding $\mathbb{Z}_{m_1}$ and…
We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the…
We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…
In this paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\mathbb{Z}_n\setminus \{0\}$ of size $k$ such that $\sum_{z\in A} z\not= 0$, it is…
Let $(a_k)_{k\in\mathbb N}$ be a sequence of integers satisfying the Hadamard gap condition $a_{k+1}/a_k>q>1$ for all $k\in\mathbb N$, and let $$ S_n(\omega) = \sum_{k=1}^n\cos(2\pi a_k \omega),\qquad n\in\mathbb N,\;\omega\in [0,1]. $$ The…
We study large deviations for random walks on Lie groups defined by $\sigma_n^n = \exp(\frac1nX_1)\cdots\exp(\frac1nX_n)$, where $\{X_n\}_{n\geq1}$ is an i.i.d sequence of bounded random variables in the Lie algebra $\mathfrak{g}$. We…
We give upper-bounds for the dimension of some linear systems. The theorem improves the differential Horace method introduced by Alexander-Hirschowitz, and was conjectured by Simpson. Possible applications are the calculus of the dimension…
We combine two of Igusa's conjectures with recent semi-continuity results by Musta\c{t}\u{a} and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in…
We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so.…
We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$.…