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Lapidus, Michel L.; van Frankenhuysen, Machiel

Complex dimensions of self-similar fractal strings and Diophantine approximation. (English) Zbl 1061.11039

Exp. Math. 12, No. 1, 41-69 (2003).
The authors study the equation \(m_1 r^s_1+\cdots+ m_d r^s_d\), where the \(r_i\)’s are given distinct positive reals and the \(m_i\)’s are given complex numbers. In the case where the \(r_i\)’s are not powers of same real number the authors show how to approximate this equation by polynomial equations of increasing degrees. They prove in particular that the solutions have a quasi-periodic structure. They also show how this question is related to complex dimensions of self-similar fractal strings [see the authors, Contemp. Math. 237, 87–105 (1999; Zbl 0945.11016) and Fractal geometry and number theory. Complex dimensions of fractal strings and zeros of zeta functions (Boston, MA, Birkhäuser) (2000; Zbl 0981.28005)].
Reviewer: Jean-Paul Allouche (Orsay)

Cited in 2 Reviews
Cited in 11 Documents

MSC:

11J99 Diophantine approximation, transcendental number theory
28A80 Fractals
11M41 Other Dirichlet series and zeta functions
37C45 Dimension theory of smooth dynamical systems
11K60 Diophantine approximation in probabilistic number theory

Keywords:

Dirichlet polynomial equations; self-similar strings; complex fractal dimension; diophantine approximation

Citations:

Zbl 0945.11016; Zbl 0981.28005
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