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For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ variables, we prove an optimal strong approximation theorem.
On the representation of numbers in the form a x 2 + b y 2 + c z 2 + d t 2 ax^{2}+by^{2}+cz^{2}+dt^{2}
H. D. Kloosterman · 1927
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The representation of a number as a sum of five or more squares (ii)
E. M. Wright · 1933
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The representation of a large number as a sum of ‘almost equal’ squares,
F. C. Auluck and S. Chowla · 1937
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The representation of a number as a sum of four almost equal squares
E. M. Wright · 1937
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Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion
Martin Eichler · 1954
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On the representation of integers by positive quadratic forms
A. V. Malyšev · 1962
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Lectures on quadratic forms
C. L. Siegel · 1967
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On the greatest prime factor of a cubic polynomial
Christopher Hooley · 1978
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Approximation of real matrices by integral matrices
R. Tijdeman · 1986
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Ramanujan graphs
A. Lubotzky, R. Phillips, and P. Sarnak · 1988
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Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators
G. A. Margulis · 1988
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Approximation of real matrices by integral matrices
Glyn Harman · 1990
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Some applications of modular forms
Peter Sarnak · 1990
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Bounds for automorphic L L -functions
W. Duke, J. Friedlander, and H. Iwaniec · 1993
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Covering with Hecke points
Patrick Chiu · 1995
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Restriction of toral eigenfunctions to hypersurfaces and nodal sets
Jean Bourgain and Zeév Rudnick · 2012
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Diophantine approximation and automorphic spectrum
Anish Ghosh, Alexander Gorodnik, and Amos Nevo · 2013
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Diophantine approximation exponents on homogeneous varieties
Anish Ghosh, Alexander Gorodnik, and Amos Nevo · 2015
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Letter to Scott Aaronson and Andy Pollington on the Solovay-Kitaev Theorem
Peter Sarnak · 2015
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Letter to Stephen D. Miller and Naser Talebizadeh Sardari on optimal strong approximation by integral points on quadrics; the case: S L 2 ( ℤ ) → S L 2 ( ℤ q ℤ ) SL_{2}(\mathbb{Z})\to SL_{2}(\frac{\mathbb{Z}}{q\mathbb{Z}})
Peter Sarnak · 2015
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D. R. Heath-Brown · 1996
Cited alongside, same era.
Functoriality for the exterior square of GL 4 {\rm GL}_{4} and the symmetric fourth of GL 2 {\rm GL}_{2}
Henry H. Kim · 2003
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Localized solutions in Waring’s problem: the lower bound
Dirk Daemen · 2010
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Anish Ghosh, Alexander Gorodnik, and Amos Nevo · 2016
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Quantum chaos on random cayley graphs of
Igor Rivin and Naser T. Sardari · 2017
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Complexity of strong approximation on the sphere
N. T Sardari · 2017
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Diameter of ramanujan graphs and random cayley graphs
Naser T. Sardari · 2018
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