On the existence of exponential polynomials with prefixed gaps

Abstract

This paper shows that the conjecture of Lapidus and Van Frankenhuysen on the set of dimensions of fractality associated with a nonlattice fractal string is true in the important special case of a generic nonlattice self-similar string, but in general is false. The proof and the counterexample of this have been given by virtue of a result on exponential polynomials P (z), with real frequencies linearly independent over the rationals, that establishes a bound for the number of gaps of RP , the closure of the set of the real projections of its zeros, and the reason for which these gaps are produced.

On the non-isolation of the real projections of the zeros of exponential polynomials

“…Now, in view of Proposition 4, we have that P * (b P ) > |P * (z 0 )| and thus, we deduce from (9) and Proposition 5 that…”

“…As far as we know, the first theoretical work on location of open intervals in R P , with P (z) an exponential polynomial of the form (1), was made by Moreno by assuming that the frequencies of P (z) are linearly independent over the rationals [11,Main Theorem] (see also [9,Theorem 1]). Later, through an auxiliary function and without specific conditions on the real frequencies of P (z), Avellar and Hale [1, Theorem 3.1] obtained a criterion to decide whether a real number is in the set R P .…”

On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

“…It is also important to highlight the following density result which can be uniquely applied to Dirichlet polynomials with real weights linearly independent over the rationals (see [10,Theorem 10]). …”

“…Thus, the results of [10] cannot be applied, except for the case n = 3, to the functions L n (z) because they have some distinct weights which are linearly dependent over the rationals (for example: log 2 and log 4). Therefore, the techniques used in this paper in order to find a new characterization of the set of points of the set R Ln := {Re z : L n (z) = 0}, for the general case, are rather different.…”

“…For the case of self-similar strings L with scaling ratios r 1 , r 2 , ..., r N and gaps g 1 , ..., g K (whose construction is reminiscent of the construction of the Cantor set), the meromorphic continuation to the whole complex plane of ζ L has the form The recent paper [10] is focused on the Dirichlet polynomials f (z) which are of the form (1.1) with weights that are linearly independent over the rationals and, hence, uniquely connected to a determined class of nonlattice fractal strings, which form the generic nonlattice self-similar case. In that paper, it was proved that the set of dimensions of fractality associated to these strings is either the entire interval that contains the projection of their complex dimensions or the union of at most n disjoint non-degenerate closed intervals.…”