Counting exceptional points for rational numbers associated to the Fibonacci sequence
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Counting exceptional points for rational numbers associated to the Fibonacci sequence Charles L. Samuels1
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract If α is a non-zero algebraic number, we let m(α) denote the Mahler measure of the minimal polynomial of α over Z. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the t-metric Mahler measure, denoted m t (α). For fixed α ∈ Q, the map t → m t (α) is continuous, and moreover, is infinitely differentiable at all but finitely many points, called exceptional points for α. It remains open to determine whether there is a sequence of elements αn ∈ Q such that the number of exceptional points for αn tends to ∞ as n → ∞. We utilize a connection with the Fibonacci sequence to formulate a conjecture on the t-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results. Keywords Mahler measure · Metric Mahler measure · Fibonacci numbers
1 Introduction Suppose α is a non-zero algebraic number with minimal polynomial over Z given by d F(z) = a · (z − αi ). i=1
Under these assumptions, the (logarithmic) Mahler measure of α is defined to be d log max{1, |αi |}. m(α) = log |a| + i=1
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Charles L. Samuels [email protected] Department of Mathematics, Christopher Newport University, 1 Avenue of the Arts, Newport News, VA 23606, USA
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C. L. Samuels ×
It is obvious from the definition that m(α) ≥ 0 for all α ∈ Q , and moreover, it follows from Kronecker’s Theorem [8] that m(α) = 0 if and only if α is a root of unity. We also note that the behavior of m(α) is particularly straightforward when α ∈ Q× . Indeed, if α = r/s and gcd(r, s) = 1 then m(α) = log max{|r |, |s|}. In attempting to construct large prime numbers, D.H. Lehmer [9] came across the problem of determining whether there exists a sequence of non-zero algebraic numbers {αn }, not roots of unity, such that m(αn ) tends to 0 as n → ∞. This problem remains unresolved, although substantial evidence suggests that no such sequence exists (see [2,10,16,18], for instance). This assertion is typically called Lehmer’s conjecture. Conjecture 1.1 (Lehmer’s Conjecture) There exists c > 0 such that m(α) ≥ c whenever × α ∈ Q is not a root of unity. Dobrowolski [4] provided the best known lower bound on m(α) in terms of deg α, while Voutier [19] later gave a version of this result with an effective constant. Nevertheless, only little progress has been made on Lehmer’s conjecture for an arbitrary algebraic number α. Dubickas and Smyth [5,6] were the first to study a modified version of the Mahler measure × × × which gives rise to a metric on Q /Qtors . A point (α1 , α2 , . . . , α N ) ∈ (Q ) N is called a N product representation of α if α = n=1 αn , and we write P (α) to denote
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