Twisted Linnik implies Optimal Covering Exponent for $S^3$

Browning, Tim; Kumaraswamy, V. Vinay; Steiner, Raphael S.

doi:10.1093/imrn/rnx116

Cited by 12 publications

(9 citation statements)

References 8 publications

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“…Remark E.3. Using [DFI93, HB96]'s delta method along the lines of [BKS19], one can likely give a heuristic argument for Conjecture E.1, assuming "generic square-root cancellation" over the modulus q (for "generic" c's). Some details, including the analysis of the oscillatory integral for small q's, would likely require some care to flesh out (cf.…”

Section: B2 Counting On Other Varieties For Simplicity We Discuss Onl...mentioning

confidence: 99%

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Diagonal cubic forms and the large sieve

Wang¹

2021

Preprint

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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] m to F (x) = 0 by O F, (X 3m/4−3/2+ ), conditionally on an "optimal large sieve inequality" (in a specific range of parameters) for "approximate Hasse-Weil L-functions" of smooth hyperplane sectionsWhen m is even, these results were previously established conditionally under Hooley's Hypothesis HW. Our 2 large sieve approach requires that certain bad factors be roughly 1 on average in 2 , while the ∞ Hypothesis HW approach only required the bound in 1 . Furthermore, the large sieve only accepts uniform vectors; yet our "initially given" vectors are only approximately uniform over c, due to variation in bad factors and in the archimedean component. Nonetheless, after some bookkeeping, partial summation, and Cauchy, the large sieve will still apply.

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Section: B2 Counting On Other Varieties For Simplicity We Discuss Onl...mentioning

confidence: 99%

“…Some details, including the analysis of the oscillatory integral for small q's, would likely require some care to flesh out (cf. [BKS19]), because k = 0.…”

Section: B2 Counting On Other Varieties For Simplicity We Discuss Onl...mentioning

confidence: 99%

Diagonal cubic forms and the large sieve

Wang¹

2021

Preprint

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“…One can establish (3.7) with exponent |S (n)| − 1 /2+ε using the 'Kloosterman circle method', see [Sar15b]. In this approach Kloosterman sums and estimates associated with them play a central role and assuming a natural conjecture about cancellations of sums involving these, one can establish that the volume covering exponent is − 3 /4 [BKS16]. That there are such points which are badly approximable by S (n) leads to a rich metric diophantine approximation theory in this and much more general contexts, see [GGN14].…”

Section: Quaternions and Ramanujanmentioning

confidence: 99%

Super-Golden-Gates for PU(2)

Parzanchevski¹,

Sarnak²

2018

Advances in Mathematics

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To each of the symmetry groups of the Platonic solids we adjoin a carefully designed involution yielding topological generators of PU (2) which have optimal covering properties as well as efficient navigation. These are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups. The generators give super efficient 1-qubit quantum gates and are natural building blocks for the design of universal quantum gates.

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“…The main goal in [3] was to show that it is possible to improve Sardari's work on covering exponents for S 3 [21] under the assumption that Conjecture 2 holds. It is unfortunate that the derived upper bounds in Theorem 1 and 3 are not strong enough to offer any unconditional improvement.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by an application to the efficiency of a certain universal set of quantum gates, Browning-Kumaraswamy-Steiner [3] have proposed the following twisted version of the Linnik-Selberg conjecture. In this paper we are concerned with establishing some progress towards this conjecture.…”

Section: Introductionmentioning

confidence: 99%