Every Infinite Achievement Set Is a Sum of Two Achievement Sets of Hausdorff Dimension Zero
A candidate proof that every achievement set generated by an absolutely convergent real series with infinitely many nonzero terms can be written as the Minkowski sum of two achievement sets 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 every infinite achievement set be decomposed into two achievable sets, each having Hausdorff dimension zero?
How the paper approaches it
Partition the inherited nonzero sequence into alternating finite blocks. Extend each block until the untouched absolute tail is super-exponentially small relative to the prefix length of the opposite factor.
What the result would establish
The candidate theorem would strengthen known zero-measure Cantor decompositions by forcing both factors to have the smallest possible nonempty Hausdorff dimension.
For a complementary partition A ⊔ B of the indices, E(a) = E(a|A) + E(a|B), while both factor achievement sets have Hausdorff 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.
Split the binary digit positions.
The explicit model assigns factorial-length blocks alternately to A and B. Their individual prefix sets are sparse, but their Minkowski sum recovers every binary prefix.
Finite prefix geometry
The dense bottom row reflects exact finite recombination. The dimension-zero conclusion depends on the infinite checkpoint construction in the manuscript.
How the argument is assembled.
The paper separates the claim into proof obligations that can be reviewed independently.
Prefix-tail cover
After m binary decisions, cover an achievement set by at most 2ᵐ translated tail sets whose diameters are controlled by the absolute remainder.
Alternating blocks
Construct complementary blocks so each factor encounters infinitely many checkpoints where its remaining tail is super-exponentially small relative to its prefix length.
Dimension zero
At the checkpoints, 2ᵐRₘˢ tends to zero for every s > 0, forcing zero Hausdorff dimension for both factors.
Exact recombination
Complementary binary selections recombine term-by-term under absolute convergence, proving the exact Minkowski-sum identity.
Exact checks and their limits.
Verification scripts test finite identities and consistency conditions. They do not replace the symbolic proofs of infinite statements.
FINITE BINARY RECOMBINATION
factorial blocks checked: 1!, 2!, 3!
digit positions: 1..9
A positions: [1, 4, 5, 6, 7, 8, 9]
B positions: [2, 3]
|C_A prefix| = 128
|C_B prefix| = 4
|C_A + C_B prefix| = 512 = 2^9
attained numerators: 0..511 over denominator 2^9
status: PASS
FACTORIAL CHECKPOINT RATIOS
r | p_r | M_(2r)/p_r | q_r | M_(2r+1)/q_r
--+-----+------------+-----+----------------
1 | 1 | 3.000000 | 2 | 4.500000
2 | 7 | 4.714286 | 26 | 5.884615
3 | 127 | 6.874016 | 746 | 7.926273
4 | 5167 | 8.947745 | 41066 | 9.962329
5 | 368047 | 10.971188 | 3669866 | 11.977198
6 | 40284847 | 12.981465 | 482671466 | 13.984620
7 | 6267305647 | 14.987025 | 87660962666 | 15.988903
8 | 1313941673647 | 16.990398 | 21010450850666 | 17.991609
9 | 357001369769647 | 18.992604 | 6423384156578666 | 19.993431
10 | 122002101778601647 | 20.994126 | 2439325392333218666 | 21.994716
lower bounds used in the paper:
M_(2r)/p_r >= r for r >= 2
M_(2r+1)/q_r >= (2r+1)/2
both checkpoint ratios diverge to infinity
status: PASS
ADAPTIVE BLOCK CHECKPOINTS FOR a_n = 2^{-n}
k=1: p_k=1, N_(2k)=2; q_k=1, N_(2k+1)=3
k=2: p_k=2, N_(2k)=4; q_k=2, N_(2k+1)=5
k=3: p_k=3, N_(2k)=9; q_k=6, N_(2k+1)=36
k=4: p_k=30, N_(2k)=900; q_k=870, N_(2k+1)=756900
every block is nonempty and both tail targets are met
status: PASS
ALL EXACT-ARITHMETIC CHECKS PASSEDFormula-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.
Tail alignment
Audit the off-by-one placement of future A and B terms after each alternating block endpoint.
Dimension criterion
Confirm the prefix-tail cover convention, arbitrarily small scales, and the checkpoint Hausdorff-content estimate.
Interpretation and prior art
Check whether the source problem imposes additional decomposition conventions and search for equivalent sparse-partition lemmas.
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). Every Infinite Achievement Set Is a Sum of Two Achievement Sets of Hausdorff Dimension Zero (Version 1.3) [Candidate research preprint]. Metriq Foundation, Inc. https://github.com/MetriqOrg/PRISM/tree/main/Papers/MF-MATH-2026-03