The variance of the error function in the shifted circle problem is a wild function of the shift (Q1319913)

The authors study the lattice rest of a convex planar domain \({\mathcal B}\) (with distance function \(f\)), ``blown up'' by a large real parameter \(R\) and shifted by some vector \({\mathbf a}= (a_ 1, a_ 2)\in \mathbb{R}^ 2\). To be precise, they assume that \(f^ 2\) is \(C^ \infty\) on \(\mathbb{R}^ 2- \{(0,0)\}\) and put \[ F_{\mathcal B} (R,{\mathbf a})= {{\#\{{\mathbf m}\in \mathbb{Z}^ 2:\;f({\mathbf m}-{\mathbf a})\leq R\}- \text{area}\{{\mathbf x}\in \mathbb{R}^ 2:\;f({\mathbf x})\leq 1\}R^ 2} \over {\sqrt{R}}}. \] In an earlier paper [Duke Math. J. 67, 461-481 (1992; Zbl 0762.11031)], the first named author had proved that the limit \[ D_{\mathcal B} ({\mathbf a})= \lim_{T\to\infty} \Biggl( {\textstyle {1\over T}} \int_ 1^ T (F_{\mathcal B} (R,{\mathbf a}))^ 2 dR\Biggr) \] always exists and is equal to \(D_{\mathcal B}({\mathbf a})= {1\over {2\pi^ 2}} \sum_{n=1}^ \infty | u_{\mathbf a}(n) |^ 2\) where \[ u_{\mathbf a}(n)=\sum _{\substack{ (k,l)\in \mathbb{Z}^ 2\\ (k,l)\neq (0,0)}} (k^ 2+ l^ 2)^{-3/4} \sqrt{\kappa (k,l)} e^{2\pi i(ka_ 1+ la_ 2)} \] and \(\kappa(k,l)\) denotes the curvature of the boundary of \({\mathcal B}\) at that point where the outward normal is collinear to \((k,l)\). In the present article the authors show that the function \(D_{\mathcal B}({\mathbf a})\) is ``wild'', i.e., continuous but nondifferentiable on a dense set of the plane: In every rational point \({\mathbf a}'\in \mathbb{Q}^ 2\), \[ \lim_{{\mathbf a}\to {\mathbf a}'} \Biggl( {{D_{\mathcal B} ({\mathbf a})- D_{\mathcal B}({\mathbf a}')} \over {|{\mathbf a}-{\mathbf a}'|}} {1\over {\log|{\mathbf a}-{\mathbf a}'|}} \Biggr)=- C_{\mathcal B}({\mathbf a}')<0. \]