# Twisted Linnik implies optimal covering exponent for $S^3$

@article{Browning2016TwistedLI, title={Twisted Linnik implies optimal covering exponent for \$S^3\$}, author={Tim D. Browning and V. Vinay Kumaraswamy and Raphael Steiner}, journal={arXiv: Number Theory}, year={2016} }

We show that a twisted variant of Linnik's conjecture on sums of Kloosterman sums leads to an optimal covering exponent for $S^3$.

## 13 Citations

ON A TWISTED VERSION OF LINNIK AND SELBERG'S CONJECTURE ON SUMS OF KLOOSTERMAN SUMS

- MathematicsMathematika
- 2019

We generalise the work of Sarnak-Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik-Selberg Conjecture.

Counting intrinsic Diophantine approximations in simple algebraic groups

- Mathematics
- 2021

We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.

Complexity of Strong Approximation on the Sphere

- Mathematics, Computer ScienceInternational Mathematics Research Notices
- 2019

It is shown that the task of accepting a number that is representable as a sum of $d\geq 2$ squares subjected to given congruence conditions is NP-complete.

N T ] 7 J an 2 02 0 PARAMETRIZATION OF KLOOSTERMAN SETS AND SL 3-KLOOSTERMAN SUMS

- Mathematics
- 2020

We stratify the SL3 big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the SL3 long word Kloosterman sum is…

Ramanujan graphs and exponential sums over function fields

- MathematicsJournal of Number Theory
- 2020

Diagonal cubic forms and the large sieve

- Mathematics
- 2021

Let F (x) be a diagonal integer-coefficient cubic form in m ∈ {4, 5, 6} variables. Excluding rational lines if m = 4, we bound the number of integral solutions x ∈ [−X,X] to F (x) = 0 by OF, (X…

Optimal strong approximation for quadratic forms

- MathematicsDuke Mathematical Journal
- 2019

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine…

Second moment of the prime geodesic theorem for $PSL_{2}(\mathbb{Z}[i])$ and bounds on a spectral exponential sum

- Mathematics
- 2019

We shall ponder the Prime Geodesic Theorem for the Picard manifold $\mathcal{M} = PSL_{2}(\mathbb{Z}[i]) \backslash \mathcal{H}^{3}$, which asks about the asymptotic behavior of the Chebyshev-like…

The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$

- Mathematics, Computer Science
- 2018

A conjecture is formulated which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb {Z}$ points of $S^{d-2}\subset S^d$.

## References

SHOWING 1-10 OF 11 REFERENCES

ON A TWISTED VERSION OF LINNIK AND SELBERG'S CONJECTURE ON SUMS OF KLOOSTERMAN SUMS

- MathematicsMathematika
- 2019

We generalise the work of Sarnak-Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik-Selberg Conjecture.

On Linnik and Selberg’s Conjecture About Sums of Kloosterman Sums

- Mathematics
- 2009

We examine the Linnik and Selberg Conjectures concerning sums of Kloosterman sums, in all its aspects (x, m and n). We correct the precise form of the Conjecture and establish an analogue of…

Simultaneous Integer Values of Pairs of Quadratic Forms

- Mathematics
- 2013

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent “almost all” pairs of integers that satisfy the necessary local conditions, provided that the…

Bounds for automorphic L-functions

- Mathematics
- 2005

on the line Re s = 2 x, the implied constant depending on s. This classical estimate resisted improvement for many years until Burgess I-B] reduced the exponent from 88 to ~ , many important…

Analytic Number Theory

- Mathematics
- 2004

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large…

Optimal strong approximation for quadratic forms

- MathematicsDuke Mathematical Journal
- 2019

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine…

A new form of the circle method, and its application to quadratic forms.

- Mathematics
- 1996

If the coefficients r(n) satisfy suitable arithmetic conditions the behaviour of F (α) will be determined by an appropriate rational approximation a/q to α, with small values of q usually producing…

Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals

- Mathematics
- 1993

PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular…

Letter to Scott Aaronson and Andy Pollington on the Solovay–Kitaev theorem

- February
- 2015

Inhomogeneous quadratic congruences

- Mathematics
- 2011

For given positive integers $a,b,q$ we investigate the density of solutions $(x,y)\in \mathbb{Z}^2$ to congruences $ax+by^2\equiv 0 \bmod{q}$.