Renormalisation of curlicues

  title={Renormalisation of curlicues},
  author={Michael V Berry and J Goldberg},
The recursively spiralling patterns drawn in the complex plane by the values of SL( tau )= Sigma Ln=1 exp(i pi tau n2) as L to infinity with tau fixed in the range 0<or= tau <or=1, depend on the arithmetic of tau . A compendious understanding of the patterns is obtained by iterating an explicit asymptotic renormalisation transformation relating SL( tau ) to a similar sum, magnified by 1/ square root tau and rotated or reflected with a smaller number L tau of terms and a new parameter tau 1( tau… Expand
The Approximate Functional Formula for the Theta Function and Diophantine Gauss Sums
We cc)nsider the polygonal lines in the complex plane C whose Nth vertex is defined by Sy = EN _o exp(iw7rn2) (with w C R), where the prime means that the first and last terms in the sum are halved.Expand
Self-similarity and growth in Birkhoff sums for the golden rotation
We study Birkhoff sums with at the golden mean rotation number with continued fraction pn/qn. The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomenaExpand
Random renormalization in the semiclassical long-time limit of a precessing spin
Abstract Discord between the semiclassical and long-time limits is illustrated by the trace of the propagator for a particle of spin J 2 = h 2 j(j+1) with Hamiltonian J2z x constant. The trace can beExpand
The Casimir effect for parallel plates revisited
The Casimir effect for a massless scalar field with Dirichlet and periodic boundary conditions (bc’s) on infinite parallel plates is revisited in the local quantum field theory (lqft) frameworkExpand
On resumming periodic orbits in the spectra of integrable systems
Spectral determinants have proved to be valuable tools for resumming the periodic orbits in the Gutzwiller trace formula of chaotic systems. We investigate these tools in the context of integrableExpand
Limiting curlicue measures for theta sums
We consider the ensemble of curves $\{\gamma_{\alpha,N}:\alpha\in(0,1],N\in\N\}$ obtained by linearly interpolating the values of the normalized theta sum $N^{-1/2}\sum_{n=0}^{N'-1}\exp(\pi iExpand
Incomplete higher order Gauss sums
Abstract We consider the classical incomplete higher-order Gauss sums S m (B)= ∑ j=0 B exp (2πij m /N), 0⩽B where N is large. In 1976, Lehmer analyzed the beautiful spirals appearing in the directedExpand
Renormalization of exponential sums and matrix cocycles
In this paper, we present a new point of view on the renormalization of some exponential sums stemming from number theory. We generalize this renormalization procedure to study some matrix cocyclesExpand
Quadratic Weyl Sums, Automorphic Functions, and Invariance Principles
Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classicalExpand
Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle
We study the one-dimensional discrete Schr\"odinger operator with the skew-shift potential $2\lambda\cos\left(2\pi \left(\binom{j}{2} \omega+jy+x\right)\right)$. This potential is long conjectured toExpand


Disorder, renormalizability, theta functions and Cornu spirals
Abstract The partial sums of the trigonometrical series S = Σ∞-∞ eiπαnp for α small mod 2 lead to distributions of points in the complex plane C composed of Cornu-like spirals. For p ≠ 2 and p > 1Expand
Quantization of linear maps on a torus-fresnel diffraction by a periodic grating
Abstract Quantization on a phase space q, p in the form of a torus (or periodized plane) with dimensions Δ q , Δ p requires the Planck's constant take one of the values h = ΔqΔp / N , where N is anExpand
Level clustering in the regular spectrum
  • M. Berry, M. Tabor
  • Mathematics
  • Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1977
In the regular spectrum of an f-dimensional system each energy level can be labelled with f quantum numbers originating in f constants of the classical motion. Levels with very different quantumExpand
Closed orbits and the regular bound spectrum
  • M. Berry, M. Tabor
  • Mathematics
  • Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1976
The energy levels of systems whose classical motion is multiply periodic are accurately given by the quantum conditions of Einstein, Brillouin & Keller (E. B. K.). We transform the E. B. K.Expand
Semiclassical approximations in wave mechanics
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with weExpand
Uniform distribution modulo one: a geometrical viewpoint.
At an early stage of our work we tacitly assumed that the curve Γ (u) generated by an equidistributed sequence u would be unbounded. This is not the case, and it seems that there are no simpleExpand
Diffraction in crystals at high energies
By means of several distinct stages of approximation, the way in which wave propagation in a lattice becomes classical at high energies is analysed. First, the principle that deflection anglesExpand
Fractal Geometry of Nature
Phase transitions in the thermodynamic formalism of multifractals.
Les non-analyticites dans les dimensions generalisees d'ensembles multifractals d'interet physique sont interpretees comme des transitions de phase
Number theory
The author briefly reviews earlier uses of number theory and then examines recent applications to music, cryptography, and error-correction codes. Expand