A thickness boundary and modular obstructions for two-set radial projections
Abstract
Let Ka,m and Kb,n be missing-digit Cantor sets with initial consecutive digit sets, and write ca=m/(a−1) and cb=n/(b−1). We prove that if ca+cb≥ 1, then the radial projection of Ka,m× Kb,n from every observer has nonempty interior; no multiplicative-independence assumption is needed for this implication. Conversely, when the bases are multiplicatively independent and ca+cb<1, we exhibit explicit unbounded open sets of observers for which the radial image is compact and nowhere dense. Rational observers with the same property are dense in each exterior corner region and occur arbitrarily close to the four corners of the product. Thus ca+cb=1 is the exact threshold for the all-observers interior property within the multiplicatively independent initial-block family. In particular, this supplies an explicit two-set counterexample to the nonempty-interior conclusions of two conjectures of Yu. We also establish a sufficient modular phase obstruction for affine translates of such sets. Combining it with a fixed-pin positive-measure theorem of Banaji and Yu yields consecutive-block division sets that are compact, perfect, of positive Lebesgue measure, and nowhere dense, with both factor dimensions tending to one. For the same family, Fourier l1-dimension estimates and the incidence argument used in Yu's product theorem imply that the two self-products contain intervals for all sufficiently large r and s, while the cross-division set remains nowhere dense.
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BibTeX
@misc{miwa2026thickness,
title = {{A thickness boundary and modular obstructions for two-set radial projections}},
author = {Yuuki Miwa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03400},
eprint = {2609.03400},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03400},
}
RIS
TY - UNPB
TI - A thickness boundary and modular obstructions for two-set radial projections
AU - Yuuki Miwa
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03400
UR - https://arxiv.org/abs/2609.03400
AB - Let K_a,m and K_b,n be missing-digit Cantor sets with initial consecutive digit sets, and write c_a=m/(a−1) and c_b=n/(b−1). We prove that if c_a+c_b≥ 1, then the radial projection of K_a,m× K_b,n from every observer has nonempty interior; no multiplicative-independence assumption is needed for this implication. Conversely, when the bases are multiplicatively independent and c_a+c_b<1, we exhibit explicit unbounded open sets of observers for which the radial image is compact and nowhere dense. Rational observers with the same property are dense in each exterior corner region and occur arbitrarily close to the four corners of the product. Thus c_a+c_b=1 is the exact threshold for the all-observers interior property within the multiplicatively independent initial-block family. In particular, this supplies an explicit two-set counterexample to the nonempty-interior conclusions of two conjectures of Yu. We also establish a sufficient modular phase obstruction for affine translates of such sets. Combining it with a fixed-pin positive-measure theorem of Banaji and Yu yields consecutive-block division sets that are compact, perfect, of positive Lebesgue measure, and nowhere dense, with both factor dimensions tending to one. For the same family, Fourier l^1-dimension estimates and the incidence argument used in Yu's product theorem imply that the two self-products contain intervals for all sufficiently large r and s, while the cross-division set remains nowhere dense.
ER -