# Renormalisation of curlicues

@article{Berry1988RenormalisationOC, title={Renormalisation of curlicues}, author={Michael V Berry and J Goldberg}, journal={Nonlinearity}, year={1988}, volume={1}, pages={1-26} }

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

#### Figures from this paper

#### 78 Citations

The Approximate Functional Formula for the Theta Function and Diophantine Gauss Sums

- Mathematics
- 1998

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

- Mathematics
- 2011

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 phenomena… Expand

Random renormalization in the semiclassical long-time limit of a precessing spin

- Physics
- 1988

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 be… Expand

The Casimir effect for parallel plates revisited

- Physics, Mathematics
- 2007

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) framework… Expand

On resumming periodic orbits in the spectra of integrable systems

- Mathematics, Physics
- 2002

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 integrable… Expand

Limiting curlicue measures for theta sums

- Mathematics
- 2009

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 i… Expand

Incomplete higher order Gauss sums

- Mathematics
- 2003

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 directed… Expand

Renormalization of exponential sums and matrix cocycles

- Art
- 2005

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 cocycles… Expand

Quadratic Weyl Sums, Automorphic Functions, and Invariance Principles

- Mathematics
- 2015

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 classical… Expand

Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle

- Mathematics, Physics
- 2018

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 to… Expand

#### References

SHOWING 1-10 OF 18 REFERENCES

Disorder, renormalizability, theta functions and Cornu spirals

- Mathematics
- 1987

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 > 1… Expand

Quantization of linear maps on a torus-fresnel diffraction by a periodic grating

- Mathematics
- 1980

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 an… Expand

Level clustering in the regular spectrum

- 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 quantum… Expand

Closed orbits and the regular bound spectrum

- 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

- Physics
- 1972

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 we… Expand

Uniform distribution modulo one: a geometrical viewpoint.

- Mathematics
- 1981

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 simple… Expand

Diffraction in crystals at high energies

- Physics
- 1971

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 angles… Expand

Phase transitions in the thermodynamic formalism of multifractals.

- Physics
- 1987

Les non-analyticites dans les dimensions generalisees d'ensembles multifractals d'interet physique sont interpretees comme des transitions de phase

Number theory

- Computer Science
- IEEE Potentials
- 1989

The author briefly reviews earlier uses of number theory and then examines recent applications to music, cryptography, and error-correction codes. Expand