An Achievable Cantorval with Zero-Dimensional Boundary
A factorial mixed-radix construction presented as a candidate affirmative answer to whether an achievable Cantorval can have boundary of Hausdorff dimension zero.
The question, construction, and claim.
This page is an interactive guide to the preprint. The manuscript and source package remain the controlling research record.
Can an achievable Cantorval have a nonempty, uncountable boundary whose Hausdorff dimension is nevertheless zero?
How the paper approaches it
Use growing even bases bₙ = 2n + 2 and blocks containing one 3/Qₙ term and n copies of 2/Qₙ. Complete residues create interior while a three-state boundary recursion controls branching.
What the result would establish
The proposed example separates topological complexity from metric size: the boundary remains a Cantor set but is dimensionally as thin as a nonempty set can be.
The proposed achievement set has nonempty interior and infinitely many gaps, while its boundary is covered at level N by at most 2·3ᴺ sets of diameter at most 2/(3Qᴺ), yielding dimension zero.
Work with the construction.
These interfaces reproduce finite structures and diagnostics described by the paper. They are explanatory tools, not substitutes for the infinite proofs.
Watch branching lose to scale.
Finite mixed-radix prefixes produce the visible set. The dimension diagnostic compares at most 2·3ᴺ boundary branches with the factorial denominator Qₙ.
Mixed-radix outer approximation
Boundary cover diagnostic
How the argument is assembled.
The paper separates the claim into proof obligations that can be reviewed independently.
Central interval
The block digit set is a complete residue system modulo bₙ. Mixed-radix prefix control yields an explicit interval inside the achievement set.
Separated branches
An exact lower-branch recursion exhibits a gap at every level, producing infinitely many complementary components and the Cantorval topology.
Boundary recursion
Track lower, central, and upper boundary states. At most three descendants survive per level, so the branch count grows like 3ᴺ.
Factorial contraction
Geometric scale contracts as Qₙ = 2ⁿ(n + 1)!, which overwhelms exponential branching. Every positive-dimensional Hausdorff cover cost tends to zero.
Exact checks and their limits.
Verification scripts test finite identities and consistency conditions. They do not replace the symbolic proofs of infinite statements.
All finite structural and high-precision checks passed.
First blocks:
n b_n Q_n block numerators D_n
1 4 4 (3,2) (0, 2, 3, 5)
2 6 24 (3,2,2) (0, 2, 3, 4, 5, 7)
3 8 192 (3,2,2,2) (0, 2, 3, 4, 5, 6, 7, 9)
4 10 1920 (3,2,2,2,2) (0, 2, 3, 4, 5, 6, 7, 8, 9, 11)
5 12 23040 (3,2,2,2,2,2) (0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13)
6 14 322560 (3,2,2,2,2,2,2) (0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15)
Closed forms:
A_1 = 2*sqrt(e)-3 = 0.297442541400256
sum E = 4*sqrt(e)-5 = 1.594885082800513
central interval = [0.594885082800513, 1]
Finite outer-approximation component counts:
U_1: 3 = 3^1
U_2: 9 = 3^2
U_3: 27 = 3^3
U_4: 81 = 3^4
U_5: 243 = 3^5
Boundary cover diagnostics:
n log(2*3^n)/log(Q_n) log10(s=0.1 cover cost)
1 1.292481250360578 0.700336125345
2 0.909478540072342 1.099642255027
3 0.758723263512023 1.486454511047
5 0.615850329442420 2.232778895882
10 0.477996864743633 3.993487849488
20 0.382582944603201 7.252951581663
50 0.297346811580000 16.015269122565
100 0.252598018043228 28.987813882235
500 0.185552261719278 110.113923820290
1,000 0.166254510903560 190.241168193062
10,000 0.123378597880982 904.120540607596
100,000 0.098036254424357 -955.736154235187Formula-derived visuals.
The source package includes the scripts and files used to generate these figures.
Where criticism is most valuable.
A useful review identifies the exact theorem, equation, inference, or prior-art issue involved and distinguishes fatal defects from repairable exposition.
Boundary recursion
Check that every displayed child is boundary of the full set, that no extra overlap points are omitted, and that central endpoints are treated correctly.
Cover count
Verify the at-most 2·3ᴺ boundary cover and its diameter bound at every level.
Prior art and generalization
Search variable-base, nonstationary Moran, and mixed-radix constructions; review the broader even-base theorem separately.
Cite the version you reviewed.
State that the result was a candidate preprint and had not undergone peer review at the time of citation.
Metriq PRISM Laboratory. (2026). An Achievable Cantorval with Zero-Dimensional Boundary (Version 1.1) [Candidate research preprint]. Metriq Foundation, Inc. https://github.com/MetriqOrg/PRISM/tree/main/Papers/MF-MATH-2026-04