The lattice point counting problem on the Heisenberg groups

Garg, Rahul; Nevo, Amos; Taylor, Krystal

doi:10.5802/aif.2986

Cited by 9 publications

(20 citation statements)

References 20 publications

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“…Upper bound, mean-square and Ω-results for E ( ): A short survey. Garg, Nevo and Taylor [9] have obtained the lower bound ≥ 2 for any value of ≥ 1. In our previous work [10], this lower bound was proved to be sharp in the case of = 1, namely 1 = 2, and so the problem is resolved for H 1 .…”

Section: Hypothesis (H)mentioning

confidence: 99%

On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups

Gath¹

2020

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A. We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan-Korányi norm ball of large radius. Let E ( ) = Z 2 +1 ∩ B − vol B 2 +2 denote the error term which occurs for this lattice point counting problem on the Heisenberg group H , where B is the unit ball in the Cygan-Korányi norm and is the Heisenberg-dilation by > 0. The characteristic behavior of the error term E ( ) may only be one of two types, depending on whether = 1 or > 1. It is the higher dimensional case that is the most challenging, and we shall confine ourselves to the case ≥ 3. To that end, for ≥ 3 we consider the suitably normalized error term E ( )/ 2 −1 , and prove it has a limiting value distribution which is absolutely continuous with respect to the Lebesgue measure. We show that the defining density for this distribution, denoted by P ( ), can be extended to the whole complex plane C as an entire function of and satisfies for any non-negative integer ≥ 0 and any ∈ R, | | > , , the bound:where > 0 is an absolute constant. In addition, we prove that ∫ ∞ −∞ P ( )d = 0 and ∫ ∞ −∞ 3 P ( )d < 0, and we give an explicit formula for the -th integral moment of the density P ( ) for any integer ≥ 1. Finally, we show that for = 1, 2, the -th integral moment of E ( )/ 2 −1 is equal to ∫ ∞ −∞ P ( )d , and more generally: lim →∞ 1 2 ∫ E ( )/ 2 −1 d = ∞ ∫ −∞ | | P ( )d for any 0 < ≤ 2.

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Section: Hypothesis (H)mentioning

confidence: 99%

On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups

Gath¹

2020

Preprint

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“…In fact, we shall first state the lattice point counting problem on an arbitrary Heisenberg group with respect to a certain family of gauges. This family, consisting of the so called Heisenberg-norms, arises naturally through the action of the dilation and unitary groups, and was considered in [7] where the reader may find an in-depth and broad treatment of the lattice point counting problem on the Heisenberg groups.…”

Section: General Counting Principlesmentioning

confidence: 99%

On the best possible exponent for the error term in the lattice point counting problem on the first Heisenberg group

Gath

2017

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We use classical methods from analytic number theory to resolve the lattice point counting problem on the first Heisenberg group, in the case where the gauge function is taken to be the Cygan-Korányi Heisenberg-norm N4,1(z, w) = (|z| 4 + w 2 ) 1/4 . In this case, our main theorem establishes the estimate E (x) = Ω±(x 1 2 ), where E (x) = S(x) − π 2 2 x is the error term arising in the lattice point counting problem, S(x) is given bym}| is the familiar sum of squares function. As a corollary, we deduce that the exponent 1 2 in the upper-bound |E (x)| ≪ x 1 2 log x obtained by Garg, Nevo & Taylor can not be improved and is thus best possible, thereby resolving the lattice point counting problem for the case in hand.

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“…In the higher dimensional case ≥ 3, the best result available to date is |E ( )| ≪ 2 −2/3 which was proved by the author ( [10], Theorem 1), and we also have ( [10], Theorem 3) the Ω-result E ( ) = Ω 2 −1 log 1/4 log log 1/8 . It follows that 8 3 ≤ ≤ 3. In regards to what should be the conjectural value of in the case of ≥ 3, it is known ( [10], Theorem 2) that E ( ) has order of magnitude 2 −1 in mean-square, which leads to the conjecture that = 3.…”

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confidence: 96%

“…. , 2 , 2 2 +1 ) with > 0 are the Heisenberg dilations, and B = u ∈ R 2 +1 : |u| ≤ 1 is the unit ball with respect to the Cygan-Korányi norm (see [8,10] for more details). It is clear that ( ) will grow for large like vol B 2 +2 , where vol(•) is the Euclidean volume, and we shall be interested in the error term resulting from this approximation.…”

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confidence: 99%

“…We shall refer to the lattice point counting problem for (2 + 1)-dimensional Cygan-Korányi balls as the problem of determining the value of . This problem was first considered by Garg, Nevo and Taylor [8], who established the lower bound ≥ 2 for all integers ≥ 1. For = 1, which is the case we shall be concerned with in the present paper, this lower bound was proven by the author to be sharp, that is 1 = 2 (see [9], Theorem 1.1, and note that a different normalization is used for the exponent of the error term).…”

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confidence: 99%

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Distribution and Moments of the Error Term in the Lattice Point Counting Problem for 3-Dimensional Cygan-Korányi Balls

Gath¹

2021

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We study fluctuations of the error term for the number of integer lattice points lying inside a 3-dimensional Cygan-Korányi ball of large radius. We prove that the error term, suitably normalized, has a limiting value distribution which is absolutely continuous, and we provide estimates for the decay rate of the corresponding density on the real line. In addition, we establish the existence of all moments for the normalized error term, and we prove that these are given by the moments of the corresponding density.

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The lattice point counting problem on the Heisenberg groups

Cited by 9 publications

References 20 publications

On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups

On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups

On the best possible exponent for the error term in the lattice point counting problem on the first Heisenberg group

Distribution and Moments of the Error Term in the Lattice Point Counting Problem for 3-Dimensional Cygan-Korányi Balls

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