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We show that a twisted variant of Linnik's conjecture on sums of Kloosterman sums leads to an optimal covering exponent for $S^3$.
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P. Sarnak and J. Tsimerman, On Linnik and Selberg’s conjecture about sums of Kloosterman sums. Algebra, Arithmetic, and Geometry
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D.R. Heath-Brown and L.B. Pierce, Simultaneous integer values of pairs of quadratic forms. J. reine angew. Math
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P. Sarnak, Letter to Scott Aaronson and Andy Pollington on the Solovay–Kitaev theorem. February, 2015
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Later among the works it cites.
E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals
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Closest in time.
R.S. Steiner, On a twisted version of Linnik and Selberg’s conjecture. In preparation
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