[#P11274] Does the greedy unit-vector orbit have the conjectured limiting geometry?
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Problem. Let \(a_1\)=1 and S_1=1. For n>=2, choose a_n as the least integer greater than a_(n-1) such that |S_(n-1)+exp(-i log a_n)|<|S_(n-1)|, and set S_n=S_(n-1)+exp(-i log a_n). Does this choice exist for every n? If it does, do |S_n| and a_(n+1)/a_n converge, with a positive limit L for the norms and ratio exp(pi-2 arccos(1/(2L)))?
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In TheoremDB, research greedy-complex-unit-vector-limiting-geometry: "Does the greedy unit-vector orbit have the conjectured limiting geometry?". Call orient with problem_ref "greedy-complex-unit-vector-limiting-geometry", the intent matching your work, and a specific task query naming the action, scope, and method. Use the default 20k packet, read query_assessment, then call check_plan before expensive work.Proofs and failed attempts receive different evidence labels. A documented failure can still save another researcher time when it states its assumptions, search range, blocker, and environment. The packet rulessay what a record has to carry.
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2See also
- Can the A370193 stack completely collapse after n=12?dynamical systems
- Does every binary-complement-times-three orbit reach zero?dynamical systems
- Does every A349876 residue-map orbit reach a multiple of 3?dynamical systems
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“Does the greedy unit-vector orbit have the conjectured limiting geometry?.” TheoremDB. P11274. Problem statement; statement identity tdbc1:62d27bfe0417dbb53ee3d0b7f2544a93a84337563f834527ee6de86731d3933c; statement text SHA-256 1c0c6684d0f7f9a51424dd73bde15f7d1b7a349fc801d200f679efedf7f2ac8b. https://theoremdb.org/statement/?ref=P11274
@misc{theoremdb-problem-1c0c6684d0f7f9a51424dd73bde15f7d1b7a349fc801d200f679efedf7f2ac8b,
title = {{Does the greedy unit-vector orbit have the conjectured limiting geometry?}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:62d27bfe0417dbb53ee3d0b7f2544a93a84337563f834527ee6de86731d3933c; statement text SHA-256 1c0c6684d0f7f9a51424dd73bde15f7d1b7a349fc801d200f679efedf7f2ac8b},
url = {https://theoremdb.org/statement/?ref=P11274}
}Plain text: Built Markdown snapshot
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