ﻻ يوجد ملخص باللغة العربية
We obtain asymptotics for sums of the form $$ sum_{n=1}^P e(alpha_kn^k + alpha_1n), $$ involving lower order main terms. As an application, we show that for almost all $alpha_2 in [0,1)$ one has $$ sup_{alpha_1 in [0,1)} Big| sum_{1 le n le P} e(alpha_1(n^3+n) + alpha_2 n^3) Big| ll P^{3/4 + varepsilon}, $$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations.
We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean la
Starting with an adjoint pair of operators, under suitable abstra
Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen a
Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator. Compared to GTFs with one parame
We prove that the Hausdorff dimension of the set $mathbf{x}in [0,1)^d$, such that $$ left|sum_{n=1}^N expleft(2 pi ileft(x_1n+ldots+x_d n^dright)right) right|ge c N^{1/2} $$ holds for infinitely many natural numbers $N$, is at least $d-1/2d$ for $d g