On a family of self-affine sets: Topology, uniqueness, simultaneous expansions
Bibliographic record
Abstract
Let $\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>1$ and $T_{i}(x,y)=((x+i)/\unicode[STIX]{x1D6FD}_{1},(y+i)/\unicode[STIX]{x1D6FD}_{2}),i\in \{\pm 1\}$ . Let $A:=A_{\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}}$ be the unique compact set satisfying $A=T_{1}(A)\cup T_{-1}(A)$ . In this paper, we give a detailed analysis of $A$ and the parameters $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$ where $A$ satisfies various topological properties. In particular, we show that if $\unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FD}_{2}<1.202$ , then $A$ has a non-empty interior, thus significantly improving the bound from Dajaniet al[Self-affine sets with positive Lebesgue measure.Indag. Math. (N.S.)25(2014), 774–784]. In the opposite direction, we prove that the connectedness locus for this family studied in Solomyak [Connectedness locus for pairs of affine maps and zeros of power series.Preprint, 2014, arXiv:1407.2563] is not simply connected. We prove that the set of points of $A$ which have a unique address has positive Hausdorff dimension for all $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$ . Finally, we investigate simultaneous $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$ -expansions of reals, which were the initial motivation for studying this family in Güntürk [Simultaneous and hybrid beta-encodings.Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference2008, pp. 743–748].
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".