[#P11560] What is the exact density of 3x-1 obstruction residues?
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Problem. For each \(L\), define \(O_L\) as follows. For an odd integer \(r>1\) with \(L>v=v_2(r-1)\), set \(m=(r-1)/2^v\), start \((a_K,b_K)=(r,2^L)\), \((a_I,b_I)=(m,2^{L-v})\), and \((c,d)=(2^{-v},-2^{-v})\). At each step set \(A_K=3a_K-1\), \(\alpha=v_2(A_K)\), \(A_I=3a_I-1\), and \(\beta=v_2(A_I)\). Stop if \(\alpha\ge v_2(b_K)\) or \(\beta\ge v_2(b_I)\). Otherwise replace \(c\) by \(c2^{\alpha-\beta}\), replace \(d\) by \((3d+c_{\mathrm{old}}-1)/2^\beta\), replace each \(a\) by the matching \(A\) divided by its power of 2, and replace each \(b\) by \(3b\) divided by that power. Put \(r\in O_L\) exactly when \(3d+c=1\) at the stopping step. The ratios \(|O_L|/2^L\) are nondecreasing and have a limit \(c_W\). Determine \(c_W\) exactly.
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“What is the exact density of 3x-1 obstruction residues?.” TheoremDB. P11560. Problem statement; statement identity tdbc1:fd4e313c38ec28cc4b7bd9e0ec9090c48779a760c94ae835f302deb231f4036d; statement text SHA-256 1a28f61dd34591086a3ce7da9440482ac9f5012f91cd9aa4baefee1cb83d3e93. https://theoremdb.org/statement/?ref=P11560
@misc{theoremdb-problem-1a28f61dd34591086a3ce7da9440482ac9f5012f91cd9aa4baefee1cb83d3e93,
title = {{What is the exact density of 3x-1 obstruction residues?}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:fd4e313c38ec28cc4b7bd9e0ec9090c48779a760c94ae835f302deb231f4036d; statement text SHA-256 1a28f61dd34591086a3ce7da9440482ac9f5012f91cd9aa4baefee1cb83d3e93},
url = {https://theoremdb.org/statement/?ref=P11560}
}Plain text: Built Markdown snapshot
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