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The 2L−12L-1 conjecture for completeness of positive linear recurrence sequences

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Let [c1,…,cL][c_1,\dots,c_L] define a positive linear recurrence sequence (PLRS) Hn\\{H_n\\}, and let BH,nB_{H,n} denote its nnth Brown's gap. The 2L−12L-1 conjecture. The sequence Hn\\{H_n\\} is complete if

BH,n≥0B_{H,n}\geq 0

for all n≤2L−1n\leq 2L-1; equivalently, Brown's criterion holds for the first 2L−12L-1 terms. A bound of 2L−12L-1 is attained by the incomplete sequence generated by [1,…,1,0,4][1,\dots,1,0,4], which first fails Brown's criterion at term 2L−12L-1. The conjecture asserts that no incomplete PLRS can first fail Brown's criterion later.

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  1. The 2L−12L-1 conjecture for completeness of positive linear recurrence sequences

    Let (Hn)(H_n) be the positive linear recurrence sequence defined by the coefficient list [c1,…,cL][c_1,\dots,c_L], and let BH,nB_{H,n} denote the quantity in Brown's criterion, so that completeness is characterized by the relevant inequalities BH,n≥0B_{H,n}\geq 0. The 2L−12L-1 conjecture. The sequence (Hn)(H_n) is complete if

    BH,n≥0for all n≤2L−1,B_{H,n}\geq 0\qquad\text{for all }n\leq 2L-1,

    that is, if Brown's criterion holds for the first 2L−12L-1 terms. This is presented as a strengthening of an earlier theorem, and no resolution is supplied in the source.

    source: Elżbieta Bołdyriew, John Haviland, Phúc Lâm, John Lentfer, Steven J. Miller and Fernando Trejos Suárez, “Completeness of Positive Linear Recurrence Sequences”, arXiv:2010.01655 (2021).

References

Primary source

Elżbieta Bołdyriew, John Haviland, Phúc Lâm, John Lentfer, Steven J. Miller and Fernando Trejos Suárez, “An Introduction to Completeness of Positive Linear Recurrence Sequences”, arXiv:2010.04071 (2020).

brown-criterioncombinatorial-number-theorylinear-recurrence-sequencespositive-linear-recurrences
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