Continuants with Equal Values, a Combinatorial Approach

Ramharter, G.; Zamboni, Luca Q.

doi:10.1007/978-3-030-85088-3_2

Cited by 2 publications

(2 citation statements)

References 5 publications

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“…In fact, since

K_n(x_1,x_2,\ldots ,x_n)=K_n(x_n,\ldots ,x_2,x_1)

, the reversal (or mirror image) of any extremal arrangement is again extremal. But more generally, the function

K(\cdot )

is far from being injective and it happens that many different permutations of the sequence y have the same K value [33]. There are many open questions concerning the distribution of the continuants

K_n(x_1,x_2,\ldots ,x_n)

(

n\in \mathbb {N})

where the

x_i

are restricted to a bounded subset of positive integers, including the famous Zaremba conjecture [37].…”

Section: Introductionmentioning

confidence: 99%

Extremal values of semi‐regular continuants and codings of interval exchange transformations

Luca

Edson²,

Zamboni

2023

Mathematika

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Given a set 𝔸 consisting of positive integers 𝑎 1 < 𝑎 2 < ⋯ < 𝑎 𝑘 and a 𝑘-term partition 𝑃 ∶ 𝑛 1 + 𝑛 2 + ⋯ + 𝑛 𝑘 = 𝑛, find the extremal denominators of the regular and semi-regular continued fraction [0; 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ] with partial quotients 𝑥 𝑖 ∈ 𝔸 and where each 𝑎 𝑖 occurs precisely 𝑛 𝑖 times in the sequence 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 . In 1983, G.Ramharter gave an explicit description of the extremal arrangements of the regular continued fraction and the minimizing arrangement for the semi-regular continued fraction and showed that in each case the arrangement is unique up to reversal and independent of the actual values of the positive integers 𝑎 𝑖 . However, an explicit determination of a maximizing arrangement for the semi-regular continuant turned out to be substantially more difficult. Ramharter conjectured that as in the other three cases, the maximizing arrangement is unique (up to reversal) and depends only on the partition 𝑃 and not on the actual values of the 𝑎 𝑖 . He further verified the conjecture in the special case of a binary alphabet. In this paper, we confirm Ramharter's conjecture for sets 𝔸 with |𝔸| = 3 and give an algorithmic procedure for constructing the unique maximizing arrangement. We also show that Ramharter's conjecture fails for sets with |𝔸| ⩾ 4 in that the maximizing

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“…In fact, since

K_n(x_1,x_2,\ldots ,x_n)=K_n(x_n,\ldots ,x_2,x_1)

, the reversal (or mirror image) of any extremal arrangement is again extremal. But more generally, the function

K(\cdot )

is far from being injective and it happens that many different permutations of the sequence y have the same K value [33]. There are many open questions concerning the distribution of the continuants

K_n(x_1,x_2,\ldots ,x_n)

(

n\in \mathbb {N})

where the

x_i

are restricted to a bounded subset of positive integers, including the famous Zaremba conjecture [37].…”

Section: Introductionmentioning

confidence: 99%

Extremal values of semi‐regular continuants and codings of interval exchange transformations

Luca

Edson²,

Zamboni

2023

Mathematika

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show abstract

“…, x 2 , x 1 ), the reversal (or mirror image) of any extremal arrangement is again extremal. But more generally, the function K(•) is far from being injective and it happens that many different permutations of the sequence y have the same K value [33]. There are many open questions concerning the distribution of the continuants K n (x 1 , x 2 , .…”

Section: Introductionmentioning

confidence: 99%

Extremal values of semi-regular continuants and codings of interval exchange transformations

Luca¹,

Edson²,

Zamboni³

2021

Preprint

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Given a set A consisting of positive integers a 1 < a 2 < • • • < a k and a k-term partition P : n 1 + n 2 + • • • + n k = n, find the extremal denominators of the regular and semi-regular continued fraction [0; x 1 , x 2 , . . . , x n ] with partial quotients x i ∈ A and where each a i occurs precisely n i times in the sequence x 1 , x 2 , . . . , x n . In 1983, G. Ramharter gave an explicit description of the extremal arrangements of the regular continued fraction and the minimizing arrangement for the semi-regular continued fraction and showed that in each case the arrangement is unique up to reversal and independent of the actual values of the positive integers a i . However, the determination of the maximizing arrangement for the semi-regular continuant turned out to be more difficult. He showed that if |A| = 2, then the maximizing arrangement is unique (up to reversal) and depends only on the partition P and not on the values of the a i . He further conjectured that this should be true for general A with |A| ≥ 2. In this paper, we confirm Ramharter's conjecture for sets A with |A| = 3 and give an algorithmic procedure for constructing the maximizing arrangement. We also show that Ramharter's conjecture fails in general for sets with |A| ≥ 4 in that the maximizing arrangement is neither unique nor independent of the values of the digits in A. The central idea, as discovered by Ramharter, is that the extremal arrangements satisfy a strong combinatorial condition. In the context of bi-infinite binary words, this condition coincides with the Markoff property, discovered by A.A. Markoff in 1879 in his study of minima of binary quadratic forms. We show that this same combinatorial condition, in the framework of infinite words over a k-letter alphabet, is the fundamental characterizing property which describes the orbit structure of codings of points under a symmetric k-interval exchange transformation.

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Continuants with Equal Values, a Combinatorial Approach

Cited by 2 publications

References 5 publications

Extremal values of semi‐regular continuants and codings of interval exchange transformations

Extremal values of semi‐regular continuants and codings of interval exchange transformations

Extremal values of semi-regular continuants and codings of interval exchange transformations

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