Renormalisation of curlicues

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Abstract
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 0L( 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 ). The authors study some special values of tau , and typical tau . Special values are: (i) tau =2/L; these are the Gauss sums, whose value (depending on L mod 4) is given exactly by their renormalisation; (ii) rational tau ; these have a finite hierarchy of curlicues; if tau =p/q then mod SL( tau ) mod grows linearly if pq is even and repeatedly retraces a finite pattern if pq is odd; (iii) quadratically irrational tau ; these are (or are attracted to) fixed points of the map tau 1( tau ); the patterns have an infinite hierarchy of curlicues self-similar under a finite number of scalings, and mod SL( tau ) mod approximately L12/; (iv) near-rational tau ; these decrease hypergeometrically under the map, with tau k+1 approximately= tau kM; their patterns are self-similar under rapidly increasing scalings and mod SL( tau ) mod approximately LM(M+1)/.

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