Abstract
A Heisenberg uniqueness pair (HUP) is a pair $(\Gamma,\Lambda)$, where $\Gamma$ is a curve in the plane and $\Lambda$ is a set in the plane, with the following property: any finite Borel measure $\mu$ in the plane supported on $\Gamma$, which is absolutely continuous with respect to arc length, and whose Fourier transform $\widehat\mu$ vanishes on $\Lambda$, must automatically be the zero measure. We prove that when $\Gamma$ is the hyperbola $x_1x_2=1$ %, and $\Lambda$ is the latticecross \[\Lambda=(\alpha\mathbb{Z}\times\{0\})\cup(\{0\}\times\beta\mathbb{Z}),\] where $\alpha,\beta$ are positive reals, then $(\Gamma,\Lambda)$ is an HUP if and only if $\alpha\beta\le1$; in this situation, the Fourier transform $\widehat\mu$ of the measure solves the onedimensional KleinGordon equation. Phrased differently, we show that \[{\mathrm e}^{\pi{\mathrm i} \alpha n t},\,\,{\mathrm e}^{\pi{\mathrm i}\beta n/t},\qquad n\in\mathbb{Z},\] span a weakstar dense subspace in $L^\infty(\mathbb{R})$ if and only if $\alpha\beta\le1$. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.

[Aa] J. Aaronson, An Introduction to Infinite Ergodic Theory, Providence, RI: Amer. Math. Soc., 1997, vol. 50.
@book {Aa, MRKEY = {1450400},
AUTHOR = {Aaronson, Jon},
TITLE = {An Introduction to Infinite Ergodic Theory},
SERIES = {Math. Surveys Monogr.},
VOLUME = {50},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1997},
PAGES = {xii+284},
ISBN = {0821804944},
MRCLASS = {28Dxx (58F11 58F17)},
MRNUMBER = {1450400},
MRREVIEWER = {Cesar E. Silva},
ZBLNUMBER = {0882.28013},
} 
[B] A. F. Beardon, The Geometry of Discrete Groups, New York: SpringerVerlag, 1983, vol. 91.
@book {B, MRKEY = {0698777},
AUTHOR = {Beardon, Alan F.},
TITLE = {The Geometry of Discrete Groups},
SERIES = {Grad. Texts in Math.},
VOLUME = {91},
PUBLISHER = {SpringerVerlag},
ADDRESS = {New York},
YEAR = {1983},
PAGES = {xii+337},
ISBN = {0387907882},
MRCLASS = {22E40 (11F06 20H15 30F35 57N10)},
MRNUMBER = {0698777},
MRREVIEWER = {Norbert Wielenberg},
ZBLNUMBER = {0528.30001},
} 
[C] F. Cellarosi, "Renewaltype limit theorem for continued fractions with even partial quotients," Ergodic Theory Dynam. Systems, vol. 29, iss. 5, pp. 14511478, 2009.
@article {C, MRKEY = {2545013},
AUTHOR = {Cellarosi, Francesco},
TITLE = {Renewaltype limit theorem for continued fractions with even partial quotients},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {29},
YEAR = {2009},
NUMBER = {5},
PAGES = {14511478},
ISSN = {01433857},
MRCLASS = {37Axx (11A55)},
MRNUMBER = {2545013},
DOI = {10.1017/S0143385708000825},
ZBLNUMBER = {1193.37013},
} 
[GM] J. Gilman and B. Maskit, "An algorithm for $2$generator Fuchsian groups," Michigan Math. J., vol. 38, iss. 1, pp. 1332, 1991.
@article {GM, MRKEY = {1091506},
AUTHOR = {Gilman, J. and Maskit, B.},
TITLE = {An algorithm for {$2$}generator {F}uchsian groups},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {The Michigan Mathematical Journal},
VOLUME = {38},
YEAR = {1991},
NUMBER = {1},
PAGES = {1332},
ISSN = {00262285},
MRCLASS = {30F35 (20H10)},
MRNUMBER = {1091506},
MRREVIEWER = {Peter J. Nicholls},
DOI = {10.1307/mmj/1029004258},
ZBLNUMBER = {0724.20033},
} 
[HJ] V. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis, New York: SpringerVerlag, 1994, vol. 28.
@book {HJ, MRKEY = {1303780},
AUTHOR = {Havin, Victor and J{ö}ricke, Burglind},
TITLE = {The Uncertainty Principle in Harmonic Analysis},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {28},
PUBLISHER = {SpringerVerlag},
ADDRESS = {New York},
YEAR = {1994},
PAGES = {xii+543},
ISBN = {354056991X},
MRCLASS = {4202 (31B35 42A16 43A45)},
MRNUMBER = {1303780},
MRREVIEWER = {Vladimir Logvinenko},
ZBLNUMBER = {0827.42001},
} 
[H] W. Heisenberg, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik," Z. Physik, vol. 43, pp. 172198, 1927.
@article{H,
author={Heisenberg, W.},
TITLE={{Ü}ber den anschaulichen {I}nhalt der quantentheoretischen {K}inematik und {M}echanik},
JOURNAL={Z. Physik},
VOLUME={43},
YEAR={1927},
PAGES={172198},
JFMNUMBER={53.0853.05},
} 
[KL] C. Kraaikamp and A. Lopes, "The theta group and the continued fraction expansion with even partial quotients," Geom. Dedicata, vol. 59, iss. 3, pp. 293333, 1996.
@article {KL, MRKEY = {1371228},
AUTHOR = {Kraaikamp, Cornelis and Lopes, Artur},
TITLE = {The theta group and the continued fraction expansion with even partial quotients},
JOURNAL = {Geom. Dedicata},
FJOURNAL = {Geometriae Dedicata},
VOLUME = {59},
YEAR = {1996},
NUMBER = {3},
PAGES = {293333},
ISSN = {00465755},
CODEN = {GEMDAT},
MRCLASS = {58F20 (11A55 58F11)},
MRNUMBER = {1371228},
DOI = {10.1007/BF00181695},
ZBLNUMBER = {0841.58049},
} 
[MP] N. Makarov and A. Poltoratski, "Meromorphic inner functions, Toeplitz kernels and the uncertainty principle," in Perspectives in Analysis, New York: SpringerVerlag, 2005, vol. 27, pp. 185252.
@incollection {MP, MRKEY = {2215727},
AUTHOR = {Makarov, N. and Poltoratski, A.},
TITLE = {Meromorphic inner functions, {T}oeplitz kernels and the uncertainty principle},
BOOKTITLE = {Perspectives in Analysis},
SERIES = {Math. Phys. Stud.},
VOLUME = {27},
PAGES = {185252},
PUBLISHER = {SpringerVerlag},
ADDRESS = {New York},
YEAR = {2005},
MRCLASS = {47B35 (30C40 46E22 47B32 47E05)},
MRNUMBER = {2215727},
MRREVIEWER = {Anton Baranov},
DOI = {10.1007/3540304347_10},
ZBLNUMBER = {1118.47020},
} 
[MS] A. L. Matheson and M. I. Stessin, "Cauchy transforms of characteristic functions and algebras generated by inner functions," Proc. Amer. Math. Soc., vol. 133, iss. 11, pp. 33613370, 2005.
@article {MS, MRKEY = {2161161},
AUTHOR = {Matheson, Alec L. and Stessin, Michael I.},
TITLE = {Cauchy transforms of characteristic functions and algebras generated by inner functions},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {133},
YEAR = {2005},
NUMBER = {11},
PAGES = {33613370},
ISSN = {00029939},
CODEN = {PAMYAR},
MRCLASS = {30D55 (30D50 30H05 46J15)},
MRNUMBER = {2161161},
MRREVIEWER = {Miroljub Jevti{ć}},
DOI = {10.1090/S000299390507913X},
ZBLNUMBER = {1087.46019},
} 
[S1] F. Schweiger, "Continued Fractions with odd and even partial quotients," Arbeitsberichte Math. Institut Universtät Salzburg, vol. 4, pp. 5970, 1982.
@article{S1,
author={Schweiger, F.},
TITLE={Continued Fractions with odd and even partial quotients},
JOURNAL={Arbeitsberichte Math. Institut Universtät Salzburg},
VOLUME={4},
YEAR={1982},
PAGES={5970},
} 
[S2] F. Schweiger, "On the approximation by continued fractions with odd and even partial quotients," Arbeitsberichte Math. Institut Universtät Salzburg, vol. 12, pp. 105114, 1984.
@article{S2,
author={Schweiger, F.},
TITLE={On the approximation by continued fractions with odd and even partial quotients},
JOURNAL={Arbeitsberichte Math. Institut Universtät Salzburg},
VOLUME={ 12},
YEAR={1984},
PAGES={105114},
}