[#P2906] Computability of the area of the Mandelbrot set

A mathematical schematic of Computability of the area of the Mandelbrot set. — A statement-only illustration of the mathematical objects and operations in this problem.

Problem. Let \(M=\{c\in\mathbb C:(z_m)_{m\ge 0}\text{ is bounded for }z_0=0,\ z_{m+1}=z_m^2+c\}\), and let \(A\) be its planar Lebesgue measure. Is \(A\) a computable real number?

1Context

Certified coverings, escape-time bounds, and parameter-space decompositions can be reused at higher precision. The target isolates the computability of one geometric invariant rather than pixelwise membership in the set.

2Definitions

Definition 1. A real number A is computable if an algorithm, given k>=1, returns a rational q with |A-q|<2^{-k}.

Definition 2. Planar Lebesgue measure is the usual two-dimensional area on C identified with R^2.

3What counts as a solution

Give an algorithm that, for every k>=1, returns a rational q with |A-q|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists.
Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled.

1Status

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow answer calls unconditional computability open and explains a conditional route through full measure of hyperbolic parameters. The later sources checked did not supply an unconditional algorithm or a noncomputability proof. Exact unresolved remainder: Give an algorithm that, for every k>=1, returns a rational q with |A-q|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled.^[2][3][1][4]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow answer calls unconditional computability open and explains a conditional route through full measure of hyperbolic parameters. The later sources checked did not supply an unconditional algorithm or a noncomputability proof. Exact unresolved remainder: Give an algorithm that, for every k>=1, returns a rational q with \|A-q\|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled.^[2][3][1][4] Relevance to this problem For Computability of the area of the Mandelbrot set, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 3, Reference 1, Reference 4 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The MathOverflow answer calls unconditional computability open and explains a conditional route through full measure of hyperbolic parameters. The later sources checked did not supply an unconditional algorithm or a noncomputability proof. Exact unresolved remainder: Give an algorithm that, for every k>=1, returns a rational q with \|A-q\|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled. Open full record →
	Trace	Reported
attempt · Route 1 Give an algorithm that, for every k>=1, returns a rational q with \|A-q\|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled.^[1] Relevance to this problem For Computability of the area of the Mandelbrot set, record mandelbrot-area-computable-next-route-20260801 (“Complete the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: Give an algorithm that, for every k>=1, returns a rational q with \|A-q\|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. Give an algorithm that, for every k>=1, returns a rational q with \|A-q\|<2^{-k}, together with a proof of its error bound, or prove that no such algorithm exists. Any numerical approximation used in a proof must include certified inner and outer area bounds whose gap is explicitly controlled. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-27. The MathOverflow answer calls unconditional computability open and explains a conditional route through full measure of hyperbolic parameters. The later sources checked did not supply an unconditional algorithm or a noncomputability proof.

Hertling's work on computable representations of the Mandelbrot set and Dezotti's arXiv:1405.1933 were checked for a result about its area. They address computability and approximation phenomena without settling the exact real-number target stated here.
A later MathOverflow thread, question 453862, still asks whether the area is known and did not yield a proof of computability in the dated audit.
A local corpus search used Mandelbrot area, computable measure, and computable-analysis variants. It found no duplicate target.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. Escape-time computation gives certified exclusion regions outside M, but a one-sided exclusion procedure alone does not compute A.

2See also

Kakeya conjecture in dimensions at least fourfractal geometry
Falconer distance conjecturefractal geometry
Planar drums whose spectra differ only finitelymathoverflow source

How to cite

TheoremDB contributors, “Computability of the area of the Mandelbrot set,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/mandelbrot-area-computable

This page as plain text: mandelbrot-area-computable.md

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1References

Packet source. MathOverflow question 249617, “Computability of the area of the Mandelbrot set,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 249617, its answer, and every visible comment were checked on 2026-07-27.Also cited at Full question, answers, and visible comments concerning Computability of the area of the Mandelbrot set; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Computability of the area of the Mandelbrot set, the reviewed source scope is Full question, answers, and visible comments concerning Computability of the area of the Mandelbrot set; checked 2026-08-01.. The packet makes no inference beyond that cited scope.Source named by the research packet.
Peter Hertling, “Is the Mandelbrot set computable?,” Mathematical Logic Quarterly 51(1) (2005), 5-18. DOI 10.1002/malq.200310124. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at main computable-analysis results and conditional criteria for the Mandelbrot set as a compact plane subset.Source used to assess the problem's recorded status.For Computability of the area of the Mandelbrot set, this source addresses computability of the set itself, a different output from computability of its Lebesgue area.
arXiv preprint 1405.1933, linked primary source for “Computability of the area of the Mandelbrot set,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:1405.1933, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract on failure of finite-sum approximation for Gronwall's filled-Julia-set area formula along the main-cardioid boundary.Source used to assess the problem's recorded status.For Computability of the area of the Mandelbrot set, this source was excluded as status evidence because it concerns individual filled Julia sets rather than computability of the Mandelbrot set's area; it is retained to document source-review history.
MathOverflow question 453862, “Computability of the area of the Mandelbrot set,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at question statement and visible answers and comments distinguishing numerical estimates from proofs of the area.Source used to assess the problem's recorded status.For Computability of the area of the Mandelbrot set, this source is a related live discussion and does not itself prove area computability.

An original CC0 textbook restatement motivated by the cited MathOverflow question; no MathOverflow prose was copied.

[#P2906] Computability of the area of the Mandelbrot set

1Context

2Definitions

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Definitions

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance