二元径向投影的厚度边界与模障碍
A thickness boundary and modular obstructions for two-set radial projections
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中文总结 AI 辅助
该研究针对缺位康托集的径向投影，确定了$c_a+c_b=1$为内部性质的阈值，构造了Yu猜想的反例，还得到了特定性质的连续块分集合及相关傅里叶维数结论。
中文摘要 AI 辅助
设$K_{a,m}$和$K_{b,n}$为具有初始连续数字集的缺位康托集，记$c_a=m/(a-1)$，$c_b=n/(b-1)$。我们证明：若$c_a+c_b\geq1$，则$K_{a,m}\times K_{b,n}$的径向投影从任意观测者视角出发都具有非空内部；该结论无需乘法独立性假设。反之，当底数为乘法独立且$c_a+c_b<1$时，我们构造了明确的无界开集观测者，其径向像为紧且无处稠密的集合；具有相同性质的有理观测者在每个外角落区域中稠密，且任意接近乘积的四个角落。因此，$c_a+c_b=1$是乘法独立初始块族内所有观测者均具有内部性质的精确阈值。特别地，这为Yu的两个猜想的非空内部结论提供了明确的二元反例。我们还建立了此类集合仿射平移的充分模相位障碍，将其与Banaji和Yu的固定锚点正测度定理结合，得到连续块分集合，该集合为紧、完备、具有正勒贝格测度且无处稠密，其两个因子维度均趋近于1。对于同一族，傅里叶$l^1$-维数估计及Yu乘积定理中使用的 incidence 论证表明，两个自乘积对所有足够大的$r$和$s$均包含区间，而交叉分集合仍保持无处稠密。
英文摘要
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 $c_a+c_b\geq 1$, then the radial projection of $K_{a,m}\times K_{b,n}$ 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.
