We study the solutions in s of a "Dirichlet polynomial equation" m 1 r s 1 + • • • + m M r s M = 1. We distinguish two cases. In the lattice case, when rj = r k j are powers of a common base r, the equation corresponds to a polynomial equation, which is readily solved numerically by using a computer. In the nonlattice case, when some ratio log rj/ log r1, j ≥ 2, is irrational, we obtain information by approximating the equation by lattice equations of higher and higher degree. We show that the set of lattice equations is dense in the set of all equations, and deduce that the roots of a nonlattice Dirichlet polynomial equation have a quasiperiodic structure, which we study in detail both theoretically and numerically.This question is connected with the study of the complex dimensions of self-similar strings. Our results suggest, in particular, that a nonlattice string possesses a set of complex dimensions with countably many real parts (fractal dimensions) which are dense in a connected interval. Moreover, we find dimension-free regions of nonlattice self-similar strings. We illustrate our theory with several examples.In the long term, this work is aimed in part at developing a Diophantine approximation theory of (higher-dimensional) selfsimilar fractals, both qualitatively and quantitatively.
Fractal geometry and number theory: complex dimensions of fractal strings and zeros of zeta functions / Michel L. Lapidus, Machiel van Frankenhuysen. p. cm. Includes bibliographical references and indexes. (acid-free paper) 1. Fractals. 2. Number theory. 3. Functions, Zeta. I. van Frankenhuysen,
We put the theory of Dirichlet series and integrals in the geometric setting of`fractal strings' (one-dimensional drums with fractal boundary). The poles of a Dirichlet series thus acquire the geometric meaning of`complex dimensions' of the associated fractal string, and they describe the geometric and spectral oscillations of this string by means of an`explicit formula'. We de ne the`spectral operator', which allows us to characterize the presence of critical zeros of zeta-functions from a large class of Dirichlet series as the question of invertibility of this operator. We thus obtain a geometric reformulation of the generalized Riemann Hypothesis, thereby extending the earlier work of the rst author with H. Maier. By considering the restriction of this operator to the subclass of`generalized Cantor strings', we prove that zeta-functions from a large subclass of this class have no in nite sequence of zeros forming a vertical arithmetic progression. (For the special case of the Riemann zeta-function, this is Putnam's theorem.) We make an extensive study of the complex dimensions of`self-similar' fractal strings, to gain further insight into the kind of geometric information contained in the complex dimensions. We also obtain a formula for the volume of the tubular neighborhoods of a fractal string and draw an analogy with Riemannian geometry. Our work suggests to de ne`fractality' as the presence of nonreal complex dimensions with positive real part.
We show that there exists an infinite sequence of sums P: a+b=c of rational integers with large height compared to the radical:-2?Âe>1.517 for l=0.5990. This improves the result of Stewart and Tijdeman [9]. The value of l comes from an asymptotic bound for the packing density of spheres. We formulate our result such that improved knowledge of l immediately improves the value of K l .
Academic PressKey Words: abc conjecture; good abc examples; the error term in the abc conjecture; sphere packing in n dimensions.
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