Mathematika最新文献

Let $$ and $$. We prove that, for any $$ and $$ as $$,

Motivated by the triangular Hilbert transform, we classify a certain family of singular Brascamp–Lieb forms which we associate with the dimension datum (1,2,2,1). We determine the exact range of Lebesgue exponents, for which one has singular Brascamp–Lieb inequalities within this family. The remaining observations concern counter examples to boundedness. We compare with a counter-example showing that the triangular Hilbert form does not satisfy singular Brascamp–Lieb bounds in the endpoints.

In this paper, we consider the flag manifold of $$ orthogonal subspaces of equal dimension that carries an action of the cyclic group of order $$. We provide a complete calculation of the associated Fadell–Husseini index. This may be thought of as an odd primary version of the computations of Baralić, Blagojevic, Karasev, and Vucic, for the Grassmann manifold $$. These results have geometric consequences for $$-fold orthogonal shadows of a convex body.

We prove upper bounds on the number of resonances and eigenvalues of Schrödinger operators $$ with complex-valued potentials, where $$ is odd. The novel feature of our upper bounds is that they are effective, in the sense that they only depend on an exponentially weighted norm of V. Our main focus is on potentials in the Lorentz space $$, but we also obtain new results for compactly supported or pointwise decaying potentials. The main technical innovation, possibly of independent interest, are singular value estimates for Fourier-extension type operators. The obtained upper bounds not only recover several known results in a unified way, they also provide new bounds for potentials that are not amenable to previous methods.

We analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape $$ for a suitable constant $$ and a polynomial $$. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $$-conjecture.

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd, and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

In this work, we estimate the sum

Let $$ and $$ be natural numbers greater or equal to 2. Let $$ be a homogeneous polynomial in $$ variables of degree $$ with integer coefficients $$, where $$ denotes the inner product, and $$ denotes the Veronese embedding with $$. Consider a variety $$ in $$, defined by $$. In this paper, we examine a set of integer vectors $$, defined by

We prove that for any $$, there exists a constant $$ such that the following is true. Let $$ be an infinite sequence of bipartite graphs such that $$ and $$ hold for all $$. Then, in any $$-edge-coloured complete graph $$, there is a collection of at most $$ monochromatic subgraphs, each of which is isomorphic to an element of $$, whose vertex sets partition $$. This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi-colour Ramsey numbers of bounded-degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.

In this article, we examine the Poissonian pair correlation (PPC) statistic for higher dimensional real sequences. Specifically, we demonstrate that for $$, almost all $$, the sequence $$ in $$ has PPC conditionally on the additive energy bound of $$. This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [Aistleitner, El-Baz, and Munsch, Geom. Funct. Anal. 31 (2021), 483–512]. More generally, we derive the PPC for $$ for almost all $$. As a consequence we establish the metric PPC for $$ provided that all of the $$ are greater than two.