Mathematika最新文献

Given a set $�A$ consisting of positive integers and a k-term partition $��P�:�n_1�+�n_2�+�\cdots �+�n_k�=�n�$, find the extremal denominators of the regular and semi-regular continued fraction $��[�0�;�x_1�,�x_2�,�\text{…}�,�x_n�]�$ with partial quotients $��x_i�\in �A�$ and where each $�a_i$

We establish a Kronecker–Weyl type result, on time-quantitative equidistribution for a natural non-integrable system, geodesic flow on the cube surface. Our tool is the shortline-ancestor method developed in Beck, Donders, and Yang [Acta Math. Hungar. 161 (2020), 66–184] and Beck, Donders, and Yang [Acta Math. Hungar. 162 (2020), 220–324], modified in an appropriate way to embrace all slopes. The method is further enhanced by the symmetry of the cube through the use of the irreducible representations of the symmetric group S₄ which makes the determination of the irregularity exponent substantially simpler.

The purpose of this work is threefold: (i) extend shadowing theory for discontinuous and non-invertible systems, (ii) consider more general classes of perturbations (for example, small only on average), (iii) establish a general theory based on the property that the shadowing holds for the case of a single perturbation. The “gluing” construction used in the analysis of the last property turns out to be the key point of this theory.

Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non-universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).

Shelah has shown that there are no chains of length ω₃ increasing modulo finite in $��{}^{{\omega }_2}�{\omega }_2�$. We improve this result to sets. That is, we show that there are no chains of length ω₃ in $�{�\left[�{\omega }_2�\right]�}^{{\aleph }_2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω₂ increasing modulo finite in $�{�\left[�{\omega }_1�\right]�}^{{\aleph }_1}$ as well as in $��{}^{{\omega }_1}�{\omega }_1�$. More generally, we study the depth of function spaces $��{}^{\kappa }�\mu �$

We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that the above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic L-function on $��G�L_m��(�m�⩾�2�)��$ and polynomial nilsequences has logarithmic decay; the other is that the mean value of the Möbius function, coefficients of automorphic L-function, and polynomial nilsequences also has logarithmic decay.

In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach and by Ellenberg and Gijswijt), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus p. Moreover, we show that for all primes $��p�\equiv �5��\mathrm{mod}��6�$ with $��p�⩽�41�$, the new construction leads to an exponentially larger growth of the affine and projective caps in $��\mathrm{AG}�(�n�,�p�)�$ and $��\mathrm{PG}�(�n�,�p�)�$. For example, when $��p�=�23�$, the existence of caps with growth $�{�\left(�8.0875�\text{…}�\right)�}^n$ follows from a three-dimensional example of Bose, and the only improvement had been to $�{�\left(�8.0901�\text{…}�\right)�}^{}$