# 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}
}
• Published 1 February 1988
• Mathematics
• Nonlinearity
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…

## Figures from this paper

• 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.
• 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
• Mathematics
Journal of High Energy Physics
• 2020
We study several quiver Chern-Simons-matter theories on the three-sphere, combining the matrix model formulation with a systematic use of Mordell’s integral, computing partition functions and
• Mathematics
• 2022
We consider quadratic Weyl sums SN (x;α, β) = ∑N n=1 exp [ 2πi (( 1 2n 2 + βn ) x+ αn )] for (α, β) ∈ Q2, where x ∈ R is randomly distributed according to a probability measure absolutely continuous
• Physics
• 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
• Mathematics
• 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
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
• Mathematics
• 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
• 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

## References

SHOWING 1-10 OF 17 REFERENCES

• Physics
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
• Physics
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.
• Mathematics
• 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
• 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
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
This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
The author briefly reviews earlier uses of number theory and then examines recent applications to music, cryptography, and error-correction codes.