Mathematika最新文献

We consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets Γ, that is, sets that contain a rotated copy of Γ in each direction. We show that for a large class of Cantor sets C and Cantor-graphs Γ built on C, the Hausdorff dimension of any Γ-Besicovitch set must be at least $$, where $$.

In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences that coincides with the usual LH for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the “generic” admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of “generic”: we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.

We prove an upper bound on the density of zeros very close to the critical line of the family of Dirichlet L-functions of modulus q at height T. To do this, we derive an asymptotic for the twisted second moment of Dirichlet L-functions uniformly in q and t. As a second application of the asymptotic formula, we prove that, for every integer q, at least 38.2% of zeros of the primitive Dirichlet L-functions of modulus q lie on the critical line.

We prove that the discrepancy of arithmetic progressions in the d-dimensional grid $�{�\{�1�,�\cdots �,�N�\}�}^d$ is within a constant factor depending only on d of $�N^{\frac{d}{�2�d�+�2�}}$. This extends the case $��d�=�1�$, which is a celebrated result of Roth and of Matoušek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valkó from about two decades ago. We further prove similarly tight bounds for grids of differing side lengths in many cases.

We derive, via the Hardy–Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of nonsingular local solubility. Our polynomials $��F��(�x�,�y�)��\in �Z��[�x_1�,�\text{…}�,�x_{s_1}�,�y_1�,�\text{…}�,�y_{s_2}�]��$ satisfy the condition that $��F��(�{\lambda }^2�x�,�\lambda �y�)��=�{\lambda }^4�F��(�x�,�y�)��$. Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in m

We present a unified approach to derive sharp isoperimetric-type inequalities of arbitrary high order. In particular, we obtain (i) sharp high-order discrete polygonal isoperimetric-type inequalities, (ii) sharp high-order isoperimetric-type inequalities for smooth curves with both upper and lower bounds for the isoperimetric deficit, and (iii) sharp higher order Chernoff-type inequalities involving a generalized width function and higher order locus of curvature centers. Our approach involves obtaining higher order discrete or smooth Wirtinger inequalities via Fourier analysis, by examining a family of linear operators. The key to our approach is identifying the appropriate linear operator and translating the analytic inequalities into geometric ones.

A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph $�G$ is the simplicial complex whose faces are the multipaths of $�G$. We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if $�G$ is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of $�G$ is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.

We provide uniform bounds for sums of Kloosterman sums in all arithmetic progressions. As a consequence, we find that Kloosterman sums do not correlate with periodic functions.

We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely solution counting in older partial progress on Vinogradov's mean value theorem corresponds to in Fourier decoupling theory.

This paper contributes to the study of a fundamental problem in the theory of quasiconformal analysis: under what conditions local quasiconformality of a homeomorphism implies its global quasisymmetry. We show that in general metric spaces local regularity and some connectivity together with the Loewner condition are necessary and sufficient for a quasiconformal map to be globally quasisymmetric with respect to the internal metrics. In this endeavor, two major new ingredients are used. One is the recently introduced concept of weakly $��(�L�,�M�)�$-quasisymmetry, serving as a bridge between local quasiconformality and global quasisymmetry. Another is the quasi-invariance of conformal modulus under weakly $��(�L�,�M�)�$-quasisymmetric maps, which is developed in this paper.