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on Bohr radius for the bidisc ($C_{59}$)

The statement

Let

\mathbb{D}^d\ :=\ \{z=(z_1,\dots,z_d)\in\mathbb{C}^d:\ \lvert z_1\rvert,\dots,\lvert z_d\rvert<1\}

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

K_d\ :=\ \sup\Bigl\{r>0:\ \lVert f_r\rVert_1\le 1\ \text{for all } f\in S_d\Bigr\}.

[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

C_{59}\ :=\ K_2,

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

0.3006\ \le\ K_2\ <\ 0.302825279492.

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

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.