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Lattice point counting in Cygan--Kor\'anyi balls on Heisenberg groups

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Pith's one-line read The lattice-point error for Cygan–Korányi balls on Heisenberg groups drops below the previous 1/3-exponent for every dimension q ≥ 4.

desk verdict Solid, modest improvement on Gath's exponent for q≥4 via a cleaner reduction to classical exponential sums; first real progress on the conjecture, nothing more. read the letter →

arxiv 2607.10971 v1 pith:FTJ3UZRE submitted 2026-07-13 math.NT math.CA

Sheng-Chen Mao , Sibei Yang This is my paper
classification math.NTmath.CA MSC 11P2143A8022E4011L03
keywords latticepointsHeisenberggroupCygan–KorányiballGausscircleproblemvanderCorputestimatesLandauformulaexponentialsums
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Counting integer points inside Cygan–Korányi balls on the Heisenberg group is the natural non-commutative analogue of the classical sphere problem. Earlier work reduced the error to size roughly t^{2q−1 + 1/3}. This paper improves the exponent for every dimension q ≥ 4 by combining a slicing formula with Landau’s classical lattice-point expansion and van der Corput’s fifth- and sixth-derivative tests. The new error is O(t^{2q−1 + 241/753}), which is the first concrete advance toward the conjectured optimal order 2q−1. For the transitional case q = 3 the same method recovers the previous bound up to a single logarithm. The argument also makes plain that the problem is essentially as hard as the two-dimensional Gauss circle problem, so further progress is expected to follow any improvement of the circle problem.

What carries the argument

A new exact formula for E_q(t) obtained by slicing the Heisenberg ball into Euclidean balls and inserting Landau’s formula for the Euclidean lattice-point error; the resulting family of 1-periodic Ψ-sums is converted by Stečkin’s inequality into exponential sums that are estimated by the simultaneous fifth- and sixth-derivative tests of van der Corput.

What would settle it

An explicit numerical computation of E_q(t) for a fixed q ≥ 4 and a sequence of large t that grows faster than t^{2q−1 + 241/753} would immediately disprove the claimed upper bound.

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Extended reading notes

Core claim

For every integer q ≥ 4 and all t ≥ 10 the lattice-point discrepancy E_q(t) of a Cygan–Korányi ball of radius t satisfies |E_q(t)| ≤ C_q t^{2q−2 + 994/753} (equivalently t^{2q−1 + 241/753}). When q = 3 the same method yields the slightly weaker bound O(t^{16/3} log t). Both estimates improve, or match up to a logarithm, the best previous results and constitute the first progress toward Gath’s conjecture that the true order is 2q−1.

Load-bearing premise

The fifth and sixth derivatives of the dual phase must stay uniformly bounded away from zero on each dyadic piece of the summation range; near the zeros of the fifth derivative this lower bound is delicate and must be checked by hand.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper studies the lattice-point error E_q(t) for Cygan–Korányi balls on the Heisenberg group H^q. After slicing and applying Landau’s formula for the Euclidean ball discrepancy in dimension 2q, the authors reduce E_q(t) to a family of 1-periodic Ψ-sums of the form ∑ Ψ(L^{-1}√(x-n^{2})+R). Stečkin’s inequality converts these into exponential sums; the B-process produces dual phases F_m whose fifth and sixth derivatives are controlled on carefully chosen dyadic pieces (including three subintervals of the critical piece U_0). Van der Corput’s 5th/6th derivative tests then yield the bound |E_q(t)| ≲ t^{2q-1+241/753} for every integer q≥4 (t≥10), while for q=3 the same argument recovers Gath’s exponent 16/3 up to a logarithmic factor. The resulting exponent is obtained by explicit balancing of the terms that arise from the derivative tests.

Significance. Gath’s conjecture asserts that the optimal order of E_q is 2q-1 for q≥3. The only previous upper bound for q≥3 was Gath’s own O(t^{2q-1+1/3}). The present work supplies the first improvement of that exponent (to 2q-1+241/753≈2q-1+0.320) for all q≥4, and does so by a comparatively elementary method that relies only on classical tools (Landau, Stečkin, van der Corput). The reduction also makes transparent the intimate link with the Gauss circle problem, suggesting that further progress on the latter (e.g., via the Bombieri–Iwaniec method) would immediately improve the Heisenberg exponent. The argument is fully explicit, free of free parameters, and self-contained.

minor comments (5)
  1. In the abstract and Theorem 1.2 the two equivalent writings of the exponent (2q-2+994/753 and 2q-1+241/753) are both correct, but a single consistent form should be chosen throughout the paper to avoid momentary confusion.
  2. Figure 1 is helpful, yet the caption and the surrounding text never state the precise range of the horizontal axis (s/m_L). Adding that information would make the zero of the fifth derivative immediately visible.
  3. After (2.11) the authors discard the case 2^{j/2}m L^{-1}<1 by a trivial bound; a one-line remark that the same bound is absorbed into the error term of the B-process would make the logic slightly cleaner.
  4. Several arXiv preprints of the first author are cited as 2026; if any have since appeared in print, the published references should be substituted.
  5. Typographical: “Derivation Test” appears once in Remark 1.3(1) instead of “Derivative Test”; “forgoing claim” on p. 9 should be “foregoing claim”.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: classical analytic derivation from Landau, Stečkin and van der Corput with no fitted parameters or load-bearing self-citation.

full rationale

The claimed bound of Theorem 1.2 is obtained by an explicit, self-contained chain: the Cygan–Korányi counting error is sliced into Euclidean ball errors (2.24), Landau’s formula (Lemma 2.1 / (2.3)) rewrites those errors as Ψ-sums, Stečkin’s inequality (Lemma 2.3) converts the Ψ-sums into exponential sums, and the 5th/6th van der Corput derivative tests (Lemma 2.7) bound the dual phases F_m after the B-process. All lower bounds on |F_m^{(5)}| and |F_m^{(6)}| are verified by direct differentiation of the explicit phase (after (2.16) and Figure 1); the final exponent 241/753 arises from ordinary term-balancing (W = L^{257/753}x^{128/753} in (2.23)). No parameter is fitted to the target quantity, no uniqueness theorem is imported from the authors’ prior work, and the only self-citations ([24], [25]) are not used as inputs to the present estimates. The derivation is therefore independent of its conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure asymptotic estimate in analytic number theory. It relies only on classical, fully proved tools (Landau's formula for Euclidean balls, van der Corput's derivative tests, Steckin's inequality, the B-process, Euler-Maclaurin). No free parameters are fitted, no new physical or geometric entities are postulated, and the only background assumptions are standard facts about the Heisenberg group and the Cygan-Koranyi norm that are recalled from the literature.

assumptions (4)
  • standard math Landau's formula for the lattice-point error of the Euclidean ball in dimension d≥4 (Lemma 2.1, taken from Landau 1924).
    Used as the starting point for the error-term expression (2.25).
  • standard math van der Corput's k-th derivative test (Lemma 2.7) and the B-process (Lemma 2.6).
    Applied to the dual phases F_m on the dyadic pieces U_j to bound the exponential sums that arise after Steckin.
  • standard math Steckin's inequality converting a Ψ-sum of bounded variation into an exponential sum (Lemma 2.3).
    Transforms the Ψ-sums that appear after Landau into objects controllable by derivative tests.
  • domain assumption The Cygan-Koranyi norm is a homogeneous gauge of degree 1 that induces a left-invariant metric on the Heisenberg group H^q.
    Standard fact recalled in the introduction; used to define the balls B_t and the error E_q(t).

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Pith. "Pith review of Lattice point counting in Cygan--Kor\'anyi balls on Heisenberg groups." pith.science (2026). https://pith.science/paper/FTJ3UZRE

@misc{pith2026260710971,
  author       = {Pith},
  title        = {Pith review of: Lattice point counting in Cygan--Kor\'anyi balls on Heisenberg groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FTJ3UZRE}},
  note         = {Machine review of arXiv:2607.10971}
}
abstract

Lattice point counting in gauge balls on the Heisenberg group $\mathbb{H}^q$ is a non-commutative analogue of the Euclidean multidimensional sphere problem, initiated by Garg, Nevo and Taylor \cite[\textit{Ann. Inst. Fourier}, 2015]{GNT15}. The case of particular interest is when the gauge is taken as the Cygan--Kor\'anyi norm and the error term reads: $$\mathcal{E}_q(t)=\#\left(\mathbb{Z}^{2 q+1} \cap \mathcal{B}_t\right)-\operatorname{vol}(\mathcal{B}_1) \, t^{2 q+2},$$ with $\mathcal{B}_t=\{(v,w)\in\mathbb{H}^q: (|v|^4 + w^2)^{1/4} \le t \}$, which is closely related to the Gauss circle problem. When $q\ge3$, Gath \cite[\textit{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 2022]{Gat22} improved upon \cite[]{GNT15} by showing that $ |\mathcal{E}_q(t)|\lesssim t^{2q-1+ 1/3}$ and proposed the conjecture that the optimal order should be $2q-1$. In this paper, through Landau's formula and the $5,6$-th Derivative Tests of van der Corput, we arrive at that $|\mathcal{E}_q(t)| \lesssim t^{2 q-1 + 241/753} $ for any $ q \geq 4$, and recover the bound of Gath for $q=3$ up to a logarithmic factor. This, via a simpler method, provides the first progress towards Gath's conjecture.

Figures

Figures reproduced from arXiv: 2607.10971 by the authors.

Figure 1
Figure 1. the 5, 6-th derivatives of x − 1 2Fm (with m = L) 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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