Mathematika最新文献

We prove that every $$-vertex linear triple system with $$ edges has at least $$ copies of a pentagon, provided $$. This provides the first nontrivial bound for a question posed by Jiang and Yepremyan. More generally, for each $$, we prove that there is a constant $$ such that if an $$-vertex graph is $$-far from being triangle-free, with $$, then it has at least $$ copies of $$. This improves the previous best bound of $$ due to Gishboliner, Shapira, and Wigderson. Our result also yields some geometric theorems, including the following. For $$ large, every $$-point set in the plane with at least $$ triangles similar to a given triangle $$, contains two triangles sharing a special point, called the harmonic point. In the other direction, we give a construction showing that the exponent $$ cannot be reduced to anything smaller than $$ and improve this further to $$ for a 3-partite version of the problem.

Szüsz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $$ under which for almost every real number $$ there exist infinitely many rationals $$ such that

The celebrated Artin conjecture on primitive roots asserts that given any integer $$ that is neither $$ nor a perfect square, there is an explicit constant $$ such that the number $$ of primes $$ for which $$ is a primitive root is asymptotically $$ as $$, where $$ counts the number of primes not exceeding $$. Artin's conjecture has remained unsolved since its formulation in 1927. Nevertheless, Hooley demonstrated in 1967 that Artin's conjecture is a consequence of the Generalized Riemann Hypothesis (GRH) for Dedekind zeta functions of certain cyclotomic-Kummer extensions over $$. In this paper, we use GRH to establish a uniform version of the Artin–Hooley asymptotic formula. Specifically, we prove that $$ whenever $$, that is, whenever $$ tends to infinity faster than any power of $$. Under GRH, we also show that the least prime $$ possessing $$ as a primitive root satisfies the upper bound $$ uniformly for all nonsquare $$. We conclude with an application to the average value of $$ and a discussion of an analog concerning the least “almost-primitive” root.

We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given their Loewner ellipsoid. For the first problem, we use the John asymmetry to unify a tight upper bound for the general case by Ball with a stronger inequality for symmetric convex bodies. We obtain an inequality that is tight for most asymmetry values in large dimensions and an even stronger inequality in the planar case that is always best possible. In contrast, we show for the second problem an inequality that is tight for bodies of any asymmetry, including cross-polytopes, parallelotopes, and (in almost all cases) simplices. Finally, we derive some consequences for the width-circumradius- and diameter-inradius-ratios when optimized over affine transformations and show connections to the Banach–Mazur distance.

We give a soft proof of a uniform upper bound for the local factors in the triple product formula, sufficient for deducing effective and general forms of quantum unique ergodicity (QUE) from subconvexity.

Let $$ be a probability measure on $$. We give conditions on the Fourier transform of its density for functionals of the form $$ to be Schur monotone. As applications, we put certain known and new results under the same umbrella, given by a condition on the Fourier transform of the density. These results include certain moment comparisons for independent and identically distributed random vectors, when the norm is given by intersection bodies, and vector-valued analogues of Khinchin's inequality with respect to appropriate norms. We also extend the discussion to higher dimensions.

Let $$ be an elliptic curve defined over $$, and let $$ be an imaginary quadratic field. Consider an odd prime $$ at which $$ has good supersingular reduction with $$ and which is inert in $$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell–Weil ranks of $$ are bounded over any subextensions of the anticyclotomic $$-extension of $$. Additionally, we provide an asymptotic formula for the growth of the $$-parts of the Tate–Shafarevich groups of $$ over these extensions.

We prove the direct and the converse inequalities for type IV superorthogonality in the vector-valued setting. The converse one is also new in the scalar setting.

We study the $$ moment of central values of the family of Dirichlet $$-functions to a fixed prime modulus and establish sharp upper bounds for all real $$.

In Euclidean space, the asymptotic shape of large cells in various types of Poisson-driven random tessellations has been the subject of a famous conjecture due to David Kendall. Since shape is a geometric concept and large cells are identified by means of geometric size functionals, the resolution of the conjecture is inevitably connected with geometric inequalities of isoperimetric type and their improvements in the form of geometric stability results, relating geometric size functionals and hitting functionals. The latter are deterministic characteristics of the underlying random tessellation. The current work explores specific and typical cells of random tessellations in spherical space. A key ingredient of our approach is new geometric inequalities and quantitative strengthenings in terms of stability results for general and also for some specific size and hitting functionals of spherically convex bodies. As a consequence, we obtain probabilistic deviation inequalities and asymptotic distributions of quite general size functionals. In contrast to the Euclidean setting, where naturally the asymptotic regime concerns large size, in the spherical framework, the asymptotic analysis is primarily concerned with high intensities.