# Journal of the Ramanujan Mathematical Society

Volume 35, Issue 2, June 2020 pp. 149–157.

The solution of the sphere problem for the Heisenberg group

**Authors**:
Yoav A. Gath

**Author institution:**Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 3200003, Israel

**Summary: **
We solve the sphere problem for the first Heisenberg group with respect to the Cygan-KoráLanyi
Heisenberg-norm. Specifically, for u = (x, y, w) ∈ ℝ ³ and t ∈ ℝ₊ let δ t (u) =
(tx, ty, t²w), N4,1(u) = ((x² + y²)² + w²)¼, and write
B4,1 t = {u ∈ ℝ ³ : N4,1(u) < t} = δt (B4,1
1) for the N4,1 ball of radius t. Let E(t) = |ℤ ³ ∩ B 4,1 t | - λ (B 4,1 t) = |ℤ ³ ∩ B 4,1 t | - π²/2
t⁴ where λ is the Lebesgue measure on ℝ ³, and set
κ = inf{0 < β < 1 : |E(t)| ≪ t⁴β}. Garg, Nevo & Taylor have established the
upper-bound |E(t)| ≪ t² log t, hence
κ ≤ ½. We shall prove that κ = ½
by showing that lim E(t)/t² = ± ∞.

Contents
Full-Text PDF