Dilated floor functions having nonnegative commutator II. Negative dilations

Lagarias, Jeffrey C.; Richman, David Harry

doi:10.4064/aa190628-14-1

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Main Results: Negative Dilations1

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“…Namely, for fixed parameters (p, q) the sporadic rational solutions are related to the non-representable positive integers in the semigroup N + p, q . We defer an treatment of this parametrization to a sequel paper [16], and refer to Ramirez-Alfonsin [20] for general facts on the Diophantine Frobenius problem.…”

Section: Main Results: Negative Dilationsmentioning

confidence: 99%

Dilated floor functions having nonnegative commutator I. Positive and mixed sign dilations

Lagarias

Richman

2019

Acta Arith.

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This paper completes the classification of the set S of all real parameter pairs (α, β) such that the dilated floor functions fα(x) = ⌊αx⌋, f β (x) = ⌊βx⌋ have a nonnegative commutator, i.e. [fα, f β ](x) = ⌊α⌊βx⌋⌋ − ⌊β⌊αx⌋⌋ ≥ 0 for all real x. It treats the negative dilation case, where both α, β < 0. This result is equivalent to classifying all positive α, β satisfying ⌊α⌈βx⌉⌋ − ⌊β⌈αx⌉⌋ ≥ 0 for all real x. The classification analysis for negative dilations is connected with the theory of Beatty sequences and with the Diophantine Frobenius problem in two dimensions.

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Section: Main Results: Negative Dilationsmentioning

confidence: 99%