Dimension conjecture for independent prime divisors

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Let C(b,D)C(b,D) be a generalized Cantor set with Hausdorff dimension cgammacgamma, with DD containing at least one of 00 and b−1b-1. Let t≥2t\geq 2 be an integer such that the sets of prime divisors of bb and tt are different. Let clambdaψclambda_{\psi} denote the approximation exponent used in the source, with clambdaψ≥1clambda_{\psi}\geq 1. Independent-prime-divisor dimension conjecture.

dim⁡H(Wt(ψ)∩C(b,D))=max⁡{1λψ+γ−1,0}.\dim_{\rm H}\bigl(W_t(\psi)\cap C(b,D)\bigr)=\max\left\{\frac{1}{\lambda_{\psi}}+\gamma-1,0\right\}.

This dimension formula is presented as a consequence of the preceding zero-full Hausdorff-measure conjecture. Thus it is conditional on that conjecture in the source and remains conjectural itself.

References

Primary source

Bing Li, Ruofan Li and Yufeng Wu, “Zero-full law for well approximable sets in missing digit sets”, arXiv:2302.03936 (2024).

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