This is the second part of a work devoted to the study of the set of integer solutions to a system of inequalities determined by homogeneous forms. According to the heuristics that the number of such solutions should match the volume of the set of real solutions, and among other results, sharp estimates for the volume of the semialgebraic domain under consideration were established in the first part. Here, these estimations are employed to establish (a set of) statements providing a global count on the number of solutions to the Diophantine inequalities under consideration. The results thus obtained go beyond the usual assumptions of smoothness and of nonvanishing of the Gaussian curvature. Specifically, the introduction of a semialgebraic level of flatness emerging from geometric tomography enables one to characterise the cases when an accurate counting estimate can be obtained in thin neighbourhoods of an algebraic variety. This is done with a specific view towards applications to Diophantine approximation.
The Lq-spectrum of a Borel measure is a fundamental concept in multifractal analysis. It is widely recognized that the Lq-spectrum associated with a fractal measure provides significant insights into its underlying dynamics and geometry. Consequently, the study of the Lq-spectrum is crucial for understanding dynamical systems and fractal measures. Our objective in this paper is to determine the exact rate of convergence of the Lq-spectra for Moran measures satisfying the Set Strong Separation Condition. As an application, we demonstrate that the empirical multifractal moment measures converge weakly to the normalized multifractal measures. Finally, we reexamine the analysis using tube formulas, and we try to show that the multifractal and fractal dimensions of the overlaps in a Moran set satisfying the Strong Open Set Condition are strictly smaller than the dimension of the set itself.
This is the second part of a work devoted to the study of the set of integer solutions to a system of inequalities determined by homogeneous forms. According to the heuristics that the number of such solutions should match the volume of the set of real solutions, and among other results, sharp estimates for the volume of the semialgebraic domain under consideration were established in the first part. Here, these estimations are employed to establish (a set of) statements providing a global count on the number of solutions to the Diophantine inequalities under consideration. The results thus obtained go beyond the usual assumptions of smoothness and of nonvanishing of the Gaussian curvature. Specifically, the introduction of a semialgebraic level of flatness emerging from geometric tomography enables one to characterise the cases when an accurate counting estimate can be obtained in thin neighbourhoods of an algebraic variety. This is done with a specific view towards applications to Diophantine approximation.
We investigate the existence and regularity of locally invariant manifolds near an approximately invariant set that satisfies a geometric hyperbolicity condition with respect to an abstract "generalized" dynamical system in Banach spaces. This hyperbolicity framework, which we term partial normal hyperbolicity, bridges the gap between normal hyperbolicity and partial hyperbolicity-concepts previously studied in finite dimensions and specific PDE contexts. Our generalized dynamical system accommodates non-smooth, non-Lipschitz, and even "non-mapping" dynamics, making it applicable to both well-posed and ill-posed differential equations. As an illustrative application, we employ our results to analyze the dynamics of whiskered tori.
We determine the maximal domain of meromorphy of any Euler product f(s1, . . . , sk) = p h(p-s1 ,...,p-sk ), where h is the quotient of two polynomials in k variables, with integer coefficients and with constant term equal to 1. More precisely, we define a domain Gamma subset of Ck, on which f admits a meromorphic extension, and such that for any z in partial derivative Gamma, there is no neighborhood of z on which f admits a meromorphic extension. This maximal domain Gamma is described as K & ring; +iRk, where K is an element of Rk is a rational cone, computed from the set of exponents of the two polynomials defining h. We also describe the divisor of f over Gamma, which comes from the local factors h(p-s1, ... , p-sk ), and from the zeta factors zeta(alpha 1s1 + & centerdot; & centerdot; & centerdot;+alpha ksk)-c alpha, where zeta denotes the Riemann zeta-function, corresponding to terms in the expansion of h as a formal infinite product li alpha(1-X1 alpha 1 & centerdot; & centerdot; & centerdot; X alpha k k )c alpha .We focus our study on the hyperplanes in the divisor, allowing us to use tools developed for the single variable case. We complete our study by giving a geometric and arithmetic description of the set of exponents occurring in the infinite product expansion of h, and by showing a new result on the geometric nature of the set of singular points of a holomorphic function defined over a tubular domain.
