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

Let $[c_1,\dots,c_L]$ define a positive linear recurrence sequence (PLRS) $\\{H_n\\}$ , and let $B_{H,n}$ denote its $n$ th Brown's gap. The $2L-1$ conjecture. The sequence $\\{H_n\\}$ is complete if

B_{H,n}\geq 0

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

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The $2L-1$ conjecture for completeness of positive linear recurrence sequences
Let $(H_n)$ be the positive linear recurrence sequence defined by the coefficient list $[c_1,\dots,c_L]$ , and let $B_{H,n}$ denote the quantity in Brown's criterion, so that completeness is characterized by the relevant inequalities $B_{H,n}\geq 0$ . The $2L-1$ conjecture. The sequence $(H_n)$ is complete if
$B_{H,n}\geq 0\qquad\text{for all }n\leq 2L-1,$
that is, if Brown's criterion holds for the first $2L-1$ 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).

Sources & referencesView supporting material

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).

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

Equivalent formulations 1