Bohr radius for the bidisc

Description of constant

Let

be the unit polydisc, and let the Schur class $S_d$ be the set of analytic functions $f:\mathbb{D}^d\to\mathbb{D}$. [Kne2025-def-polydisc] [Kne2025-def-Schur]

Writing the power series expansion $f(z)=\sum_{\alpha\in\mathbb{N}_0^d} f_\alpha z^\alpha$, define the coefficient-wise $\ell^1$ norm $\lVert f\rVert_1:=\sum_\alpha \lvert f_\alpha\rvert$ and the dilation $f_r(z):=f(rz)$. [Kne2025-def-l1] [Kne2025-def-fr]

The Bohr radius $K_d$ is defined by

[Kne2025-def-Kd]

Equivalently, $K_d$ is the largest number such that for every power series $\sum_\alpha c_\alpha z^\alpha$ with $\bigl\lvert\sum_\alpha c_\alpha z^\alpha\bigr\rvert<1$ on $\mathbb{D}^d$, one has $\sum_\alpha \lvert c_\alpha z^\alpha\rvert<1$ whenever $\max_{1\le j\le d}\lvert z_j\rvert<K_d$. [BK1997-def-Kn]

We define

the Bohr radius for the bidisc $\mathbb{D}^2$.

Bohr’s one-variable theorem gives $K_1=1/3$, and in particular implies $K_2\le 1/3$. [BK1997-Bohr-1d] [BK1997-ub-1-3]

The exact value of $K_d$ is unknown for every $d>1$; in particular, the exact value of $K_2$ is open. [BK1997-open]

The best established range currently is

[Kne2025-lb-K2-0-3006] [P2026-ub-K2-0-302825279492]

Known upper bounds

Known lower bounds

Additional comments and links

• Asymptotics in high dimension. The Bohr radius satisfies $K_d\asymp \sqrt{(\log d)/d}$ as $d\to\infty$ (up to absolute constants). [Kne2025-asymp-Kd]

References

• [BK1997] Boas, Harold P.; Khavinson, Dmitry. Bohr’s power series theorem in several variables. Proceedings of the American Mathematical Society 125 (1997), no. 10, 2975–2979. DOI: https://doi.org/10.1090/S0002-9939-97-04270-6. arXiv PDF: https://arxiv.org/pdf/math/9606203. Google Scholar
  • [BK1997-Bohr-1d] loc: arXiv v1 PDF p.1, Theorem 1 quote: “Then $\sum_{k=0}^\infty \lvert c_k z^k\rvert<1$ when $\lvert z\rvert<1/3$. Moreover, the radius $1/3$ is the best possible.”
  • [BK1997-def-Kn] loc: arXiv v1 PDF p.1, definition paragraph for $K_n$ quote: “Let $K_n$ denote the n-dimensional Bohr radius: the largest number such that if $\sum_\alpha c_\alpha z^\alpha$ converges in the unit polydisc $\{(z_1,\ldots,z_n):\max_{1\le j\le n}\lvert z_j\rvert<1\}$, and if $\left\lvert\sum_\alpha c_\alpha z^\alpha\right\rvert<1$ in the unit polydisc, then $\sum_\alpha \lvert c_\alpha z^\alpha\rvert<1$ when $\max_{1\le j\le n}\lvert z_j\rvert<K_n$.”
  • [BK1997-ub-1-3] loc: arXiv v1 PDF p.2, paragraph after definition of $K_n$ quote: “It is evident from Bohr’s one-dimensional result that $K_n\le 1/3$ for every $n$.”
  • [BK1997-lb-1-3sqrt] loc: arXiv v1 PDF p.2, Proof of Theorem 2 quote: “This ball evidently contains the polydisc $\{z:\max_{1\le j\le n}\lvert z_j\rvert<1/(3\sqrt{n})\}$, whence $K_n\ge 1/(3\sqrt{n})$.”
  • [BK1997-open] loc: arXiv v1 PDF p.2, Open question quote: “Open question. What is the exact value of the Bohr radius $K_n$ when $n>1$?”
• [Kne2025] Knese, Greg. Three radii associated to Schur functions on the polydisk. Proceedings of the American Mathematical Society, Series B 12 (2025), no. 5, 48–63. DOI: https://doi.org/10.1090/bproc/262. arXiv PDF: https://arxiv.org/pdf/2410.21693. Google Scholar
  • [Kne2025-def-polydisc] loc: arXiv v3 PDF p.1, Introduction quote: “$\mathbb{D}^d=\{z=(z_1,\dots,z_d)\in\mathbb{C}^d:\ \lvert z_1\rvert,\dots,\lvert z_d\rvert<1\}$.”
  • [Kne2025-def-Schur] loc: arXiv v3 PDF p.1, Introduction quote: “The Schur class $S_d$ of the polydisk $\mathbb{D}^d$ is the set of all analytic $f:\mathbb{D}^d\to\mathbb{D}$.”
  • [Kne2025-def-l1] loc: arXiv v3 PDF p.1, equation (1.1) context in Introduction quote: “Define the coefficient-wise $\ell^1$ norm $\lVert f\rVert_1:=\sum_\alpha \lvert f_\alpha\rvert$.”
  • [Kne2025-def-fr] loc: arXiv v3 PDF p.1, Introduction quote: “For $r>0$ define $f_r(z):=f(rz)$.”
  • [Kne2025-def-Kd] loc: arXiv v3 PDF p.2, Introduction quote: “Define the Bohr radius $K_d$ by $K_d:=\sup\{r>0:\ \lVert f_r\rVert_1\le 1\ \text{for all } f\in S_d\}$.”
  • [Kne2025-lb-K2-0-3006] loc: arXiv v3 PDF, Corollary 1.2 quote: “Corollary 1.2. $K(\mathcal{A}_2)=K_2\ge 0.3006$.”
  • [Kne2025-asymp-Kd] loc: arXiv v3 PDF p.2, Introduction quote: “After the culmination of deep work by many authors the precise asymptotic $K_d\sim \sqrt{\log d/d}$ was established; see [18], [8].”
• [BPWW2026] Baran, Radomił; Pikul, Piotr; Woerdeman, Hugo J.; Wojtylak, Michał. Contractive realization theory for the annulus and other intersections of disks on the Riemann sphere. Journal of Functional Analysis 290 (2026), no. 8, 111346. DOI: https://doi.org/10.1016/j.jfa.2026.111346. arXiv PDF: https://arxiv.org/pdf/2504.03236. Google Scholar
  • [BPWW2026-known-interval] loc: arXiv v1 PDF p.2, Introduction quote: “The constant $K_2$ is the 2-variate version of the Bohr constant, and is known to lie in the interval $(0.3006,1/3)$. We are able to narrow the interval to $(0.3006,0.3177)$ in Theorem 6.4.”
  • [BPWW2026-ub-K2-0-3177] loc: arXiv v1 PDF p.18, Theorem 6.4 quote: “Theorem 6.4. $K_2<0.3177$.”
• [G2026] Griego, Sebastian. Rational-inner Fejer averaging certificate for the bidisc Bohr radius bound $K_2<0.3174541$, submitted to this repository (2026).
  • [G2026-ub-K2-0-3174541] loc: pull request certificate and exact verifier quote: “The exact integer verifier proves $B_{3174541/10000000}(p)>1+1/7307638490$ and the rational-inner Fejer averaging certificate proves $\lvert p\rvert\le 1$ on the bidisc.”
• [P2026] Patel, Shivam. Complex phases and a certified upper bound for the bidisc Bohr radius. Research note and reproducibility record, 26 August 2026. MathDB solution. Exact Python and Lean proof package, v1.0.0, Zenodo, 5 September 2026, DOI: 10.5281/zenodo.22341928; GitHub release; pinned source.
  • [P2026-ub-K2-0-302825279492] loc: the analytic and finite coefficient certificates in the package README; certificate/EndToEnd.lean, theorem Optim.BohrRadius.bohrRadius_lt_302825279492_div_10pow12 statement: The explicit Schur witness and exact finite coefficient comparison prove $K_2<302825279492/10^{12}=0.302825279492$. The exact value of $K_2$ remains open.

Contribution notes

Prepared initially with assistance from ChatGPT 5.2 Pro and updated with assistance from ChatGPT 5.5 Pro. The Patel update, independent exact verifiers, and Lean formalization were prepared with assistance from Codex. Shivam Patel reviewed the original references and submitted information. The external packaging and Markdown revision were prepared with Codex assistance.

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