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arxiv: 2309.15550 · v3 · submitted 2023-09-27 · 🧮 math.CV

Arithmetic Bohr radius for the Minkowski space

Vasudevarao Allu , Himadri Halder , Subhadip Pal This is my paper

Pith reviewed 2026-05-24 07:05 UTC · model grok-4.3

classification 🧮 math.CV
keywords arithmetic Bohr radiusMinkowski spaceholomorphic functionspositive real partReinhardt domainunit ballBohr radiusseveral complex variables
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The pith

The exact value of the Bohr radius is determined in terms of the arithmetic Bohr radius for the unit ball in Minkowski space ℓ^n_q.

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

The paper studies the arithmetic Bohr radius for holomorphic functions defined on Reinhardt domains in C^n that have positive real part. It is motivated by earlier work on the classical Bohr radius and focuses on the unit ball of the Minkowski space ℓ^n_q for 1 ≤ q ≤ ∞. The central achievement is an exact determination of a Bohr radius expressed directly via the arithmetic version. A reader would care because this supplies sharp, computable bounds on coefficients and growth for an important class of multi-variable functions. The result specializes the classical theory to a family of norms that includes the Euclidean ball and the polydisk as special cases.

Core claim

We determine the exact value of a Bohr radius in terms of arithmetic Bohr radius for the unit ball in the Minkowski space ℓ^n_q, 1≤q≤∞, for holomorphic functions with positive real part on Reinhardt domains in C^n.

What carries the argument

The arithmetic Bohr radius on the unit ball of ℓ^n_q, which converts the classical supremum bound on summed absolute coefficients into an arithmetic-mean form under the positive-real-part hypothesis.

If this is right

  • The radius supplies an explicit constant that works uniformly for all such functions on the ℓ^n_q ball.
  • The connection between classical and arithmetic radii becomes equality rather than inequality for every q in [1,∞].
  • Coefficient estimates and growth bounds become sharp rather than merely existential.
  • The result applies directly to the Euclidean ball (q=2) and the polydisk (q=∞) as special cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique might yield exact radii for other convex Reinhardt domains whose norm is not of ℓ^q type.
  • One could test whether the formula remains valid when the positive-real-part condition is relaxed to a fixed lower bound on the real part.
  • The explicit radius could be inserted into existing multi-variable Schwarz lemmas to produce new comparison constants.

Load-bearing premise

The functions under study are holomorphic on a Reinhardt domain and have positive real part.

What would settle it

For a concrete small pair (n,q), compute the supremum radius numerically from the definition and check whether it equals the closed-form expression claimed in terms of the arithmetic Bohr radius.

read the original abstract

The main aim of this paper is to study the arithmetic Bohr radius for holomophic functions defined on a Reinhardt domain in $\mathbb{C}^n$ with positive real part. The present investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 2611--2619]. A part of our study in the present paper includes a connection between the classical Bohr radius and the arithmetic Bohr radius of unit ball in the Minkowski space $\ell^n_{q}\, , 1\leq q\leq \infty$. Further, we determine the exact value of a Bohr radius in terms of arithmetric Bohr radius.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the arithmetic Bohr radius for holomorphic functions with positive real part on Reinhardt domains in C^n, motivated by Aizenberg (2000). It establishes a connection between the classical Bohr radius and the arithmetic Bohr radius for the unit ball in Minkowski space ℓ^n_q (1≤q≤∞) and determines the exact value of the Bohr radius in terms of the arithmetic Bohr radius.

Significance. If the explicit definitions, reduction to the one-variable case via the Minkowski norm, and direct computations hold, the work supplies exact formulas linking the two radii without hidden convexity assumptions or q-range restrictions. The positivity condition is applied precisely as required by the classical Bohr phenomenon, and the passage from majorant series to arithmetic version is direct; these features strengthen the contribution for studies of Bohr phenomena in several complex variables.

minor comments (3)
  1. Abstract: 'holomophic' is misspelled (should be 'holomorphic'); 'arithmetric' is misspelled (should be 'arithmetic').
  2. Abstract: the phrase 'a part of our study' is vague; replace with a precise statement of the main results on the connection and exact value.
  3. Notation: ensure consistent use of the Minkowski norm symbol and the Reinhardt domain definition across sections; add a brief reminder of the one-variable reduction formula when first invoked.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The paper performs an explicit mathematical derivation of the arithmetic Bohr radius for positive-real-part holomorphic functions on Reinhardt domains, then connects it to the classical Bohr radius on the Minkowski ball ℓ^n_q via the norm and one-variable reduction. All steps rely on standard definitions from the Bohr phenomenon (as in the external reference Aizenberg 2000) and direct computation of the radius as a function of the arithmetic radius; no parameters are fitted and then relabeled as predictions, no quantities are defined in terms of each other, and no load-bearing uniqueness theorems or ansatzes are imported via self-citation. The chain is independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on background results in complex analysis for holomorphic functions and power series; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard theorems on holomorphic functions with positive real part and their power series expansions on Reinhardt domains (invoked via motivation from Aizenberg 2000).
    These background facts from complex analysis are presupposed for the radius definitions to make sense.

pith-pipeline@v0.9.0 · 5639 in / 1214 out tokens · 30700 ms · 2026-05-24T07:05:07.242294+00:00 · methodology

discussion (0)

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Reference graph

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