Filtered rays over iterated absolute differences on layers of integers

The dynamical system generated by the iterated calculation of the high order gaps between neighboring terms of a sequence of natural numbers is remarkable and only incidentally characterized at the boundary by the notable Proth-Glibreath Conjecture for prime numbers. We introduce a natural extension of the original triangular arrangement, obtaining a growing hexagonal covering of the plane. This is just the base level of what further becomes an endless discrete helicoidal surface. % Although the repeated calculation of higher-order gaps causes the numbers that generate the helicoidal surface to decrease, there is no guarantee, and most often it does not even happen, that the levels of the helicoid have any regularity, at least at the bottom levels. However, we prove that there exists a large and nontrivial class of sequences with the property that their helicoids have all levels coinciding with their base levels. This class includes in particular many ultimately binary sequences with a special header. % For almost all of these sequences, we additionally show that although the patterns generated by them seem to fall somewhere between ordered and disordered, exhibiting fractal-like and random qualities at the same time, the distribution of zero and non-zero numbers at the base level has uniformity characteristics. Thus, we prove that a multitude of straight lines that traverse the patterns encounter zero and non-zero numbers in almost equal proportions

Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics

We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasi-momentum space the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider net in contrast to the Hofstadter butterfly for unbiased motion.Comment: 12 pages, 18 figures. Fractal properties of the energy levels are visualised in the supplementary video material https://www.youtube.com/watch?v=ODS3QVkPTP

Quantumlike Chaos in the Frequency Distributions of the Bases A, C, G, T in Drosophila DNA

Continuous periodogram power spectral analyses of fractal fluctuations of frequency distributions of bases A, C, G, T in Drosophila DNA show that the power spectra follow the universal inverse power-law form of the statistical normal distribution. Inverse power-law form for power spectra of space-time fluctuations is generic to dynamical systems in nature and is identified as self-organized criticality. The author has developed a general systems theory, which provides universal quantification for observed self-organized criticality in terms of the statistical normal distribution. The long-range correlations intrinsic to self-organized criticality in macro-scale dynamical systems are a signature of quantumlike chaos. The fractal fluctuations self-organize to form an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. Power spectral analysis resolves such a spiral trajectory as an eddy continuum with embedded dominant wavebands. The dominant peak periodicities are functions of the golden mean. The observed fractal frequency distributions of the Drosophila DNA base sequences exhibit quasicrystalline structure with long-range spatial correlations or self-organized criticality. Modification of the DNA base sequence structure at any location may have significant noticeable effects on the function of the DNA molecule as a whole. The presence of non-coding introns may not be redundant, but serve to organize the effective functioning of the coding exons in the DNA molecule as a complete unit.Comment: 46 pages, 9 figure

On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated, and characterized up to a measure zero set, by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of another conjecture that we call Aubry-Godr\`eche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyri\`ere and M. Queff\'elec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.Comment: 35 pages In honor of the 60-th birthday of Henri Cohe