Irrationality Exponent Research Articles

On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values

We consider the behavior of the power series $\mathfrak{M}_0(z)=\sum_{n=1}^{\infty}\mu^2(n)z^n$ as $z$ tends to $e(\beta)=e^{2\pi i\beta}$ along a radius of the unit circle. If $\beta$ is irrational with irrationality exponent 2 then $\mathfrak{M}_0(e(\be

Hankel determinants of the Cantor sequence

In the paper, we give the recurrent equations of the Hankel determinants of the Cantor sequence, and show that the Hankel determinants as a double sequence is 3-automatic. With the help of the Hankel determinants, we prove that the irrationality exponent of the Cantor number, i.e. the transcendental number with Cantor sequence as its b-ary expansion, equals 2.

On simultaneous diophantine approximations to [formula omitted] and [formula omitted

We present a hypergeometric construction of rational approximations to ζ(2) and ζ(3) which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, ζ(2) and ζ(3) with rational coefficients. We then go further to formalize the arithmetic structure of these specific linear forms by introducing a new notion of (simultaneous) diophantine exponent. Finally, we study the properties of this newer concept and link it to the classical irrationality exponent and its generalizations given recently by S. Fischler.

Two hypergeometric tales and a new irrationality measure of $$\zeta (2)$$ ζ ( 2 )

We prove the new upper bound 5.095412 for the irrationality exponent of $\zeta(2)=\pi^2/6$; the earlier record bound 5.441243 was established in 1996 by G. Rhin and C. Viola.

On the irrationality exponent of the regular paperfolding numbers

In this paper, improving the method of Allouche et al., we calculate the Hankel determinant of the regular paperfolding sequence, and prove that the Hankel determinant sequence modulo 2 is periodic with period 10 which answers Coonʼs conjecture. Then we extend Bugeaudʼs method to obatin the exact value of the irrationality exponent for some general transcendental numbers. Using the results above, we prove that the irrationality exponents of the regular paperfolding numbers are exactly 2.

Quadratic irrationality exponents of certain numbers

The paper presents upper estimates for the non-quadraticity measure of the numbers $\sqrt {2k + 1} \ln ((k + 1 - \sqrt {2k + 1} /k)$ and $\sqrt {2k - 1} arctg(\sqrt {2k - 1} /(k - 1))$ , where k ∈ ℕ. In particular, the upper estimate for the non-quadraticity measure of ln 2 is improved.

Rational approximations to algebraic Laurent series with coefficients in a finite field

We give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α−p/q|<q −μ has infinitely many solutions (p,q)∈ℤ×ℕ. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n⩾1. We then use this result to prove that for b⩾2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.

Scaling Laws for the Response of Nonlinear Elastic Media with Implications for Cell Mechanics

We show how strain stiffening affects the elastic response to internal forces, caused either by material defects and inhomogeneities or by active forces that molecular motors generate in living cells. For a spherical force dipole in a material with a strongly nonlinear strain energy density, strains change sign with distance, indicating that, even around a contractile inclusion or molecular motor, there is radial compression; it is only at a long distance that one recovers the linear response in which the medium is radially stretched. Scaling laws with irrational exponents relate the far-field renormalized strain to the near-field strain applied by the inclusion or active force.

Square exponent of irrationality of ln2

DOI: 10.3103/S0027132212010044 For each number α that is not a root of a square equation with integer coefficients, one can introduce the characteristic of how accurate it can be approximated by roots of square equations. This characteristic called the square exponent of irrationality of the number α is defined as the supremum of the set of numbers κ such that the inequality |α− β| 4a, and the polynomial

Nesterenko’s criterion when the small linear forms oscillate

In this paper we generalize Nesterenko’s criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to ζ(5), ζ(7), ζ(9), and ζ(11), using Zudilin’s proof that at least one of these numbers is irrational.

The Euclid Algorithm is totally gaussian

We consider Euclid’s gcd algorithm for two integers $(p, q)$ with $1 \leq p \leq q \leq N$, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function $d$ on the set of possible digits, asymptotically for $N \to \infty$. For any additive cost of moderate growth $d$, Baladi and Vallée obtained a central limit theorem, and –in the case when the cost $d$ is lattice– a local limit theorem. In both cases, the optimal speed was attained. When the cost is non lattice, the problem was later considered by Baladi and Hachemi, who obtained a local limit theorem under an intertwined diophantine condition which involves the cost $d$ together with the “canonical” cost $c$ of the underlying dynamical system. The speed depends on the irrationality exponent that intervenes in the diophantine condition. We show here how to replace this diophantine condition by another diophantine condition, much more natural, which already intervenes in simpler problems of the same vein, and only involves the cost $d$. This “replacement” is made possible by using the additivity of cost $d$, together with a strong property satisfied by the Euclidean Dynamical System, which states that the cost $c$ is both “strongly” non additive and diophantine in a precise sense. We thus obtain a local limit theorem, whose speed is related to the irrationality exponent which intervenes in the new diophantine condition. We mainly use the previous proof of Baladi and Hachemi, and “just” explain how their diophantine condition may be replaced by our condition. Our result also provides a precise comparison between the rational trajectories of the Euclid dynamical system and the real trajectories.

Morphic and automatic words: maximal blocks and Diophantine approximation

Let w be a morphic word over a finite alphabet �, and letbe a nonempty subset of �. We study the behavior of maximal blocks consisting only of letters fromin w, and prove the following: let (ik, jk) denote the starting and ending positions, respectively, of the k'th maximal �-block in w. Then limsupk!1(jk/ik) is algebraic if w is morphic, and rational if w is automatic. As a result, we show that the same conclusion holds if (ik, jk) are the starting and ending positions of the k'th maximal zero block, and, more generally, of the k'th maximal x- block, where x is an arbitrary word. This enables us to draw conclusions about the irrationality exponent of automatic and morphic numbers. In particular, we show that the irrationality exponent of automatic (resp., morphic) numbers belonging to a certain class that we define is rational (resp., algebraic).

Fermat's Last Theorem for Fractional and Irrational Exponents

SummaryFermat's Last Theorem says that for integers n ≥ 2, there are no solutions to xn + yn = zn among positive integers. What about rational exponents n? Irrational n? Negative n? See what an undergraduate senior seminar discovered.

Exponents of Diophantine approximation and expansions in integer bases

Let ξ be a real number and let b ⩾ 2 be an integer. Let vb(ξ) or v′b(ξ) denote the supremum of the real numbers v for which the equation ‖bn ξ‖ ⩽ (bn)−v or ‖br (bs−1) ξ‖ ⩽ (br+s)−v has infinitely many solutions in positive integers n or r and s, respectively. Here, ‖·‖ stands for the distance to the nearest integer. Also let v1 (ξ) denote the supremum of the real numbers v for which the equation ‖q ξ‖ < q−v has infinitely many solutions in positive integers q. Motivated by the question whether one can read the irrationality exponent of a real number on its b-ary expansion, we establish various results on the set of values taken by the triple of functions (v1, vb, v′b) evaluated at real points.

Extremality Properties of Some Diophantine Series

We study the convergence properties of the series Ψ s (α) := with respect to the values of the real numbers α and s, where ||x|| is the distance of x to . For example, when s ∊ (0, 1], the convergence of Ψ s (α) strongly depends on the Diophantine nature of α, mainly its irrationality exponent. We also conjecture that Ψ s (α) is minimal at √5 for s ∊ (0, 1], and we present evidence in favor of that conjecture. For s = 1, we formulate a more precise conjecture about the value of the abscissa uk where the Fk -partial sum of Ψ1(α) is minimal, Fk being the kth Fibonacci number. A similar study is made for the partial sums of the series , which we conjecture to be minimal at √2/2.

Irrationality exponent and rational approximations with prescribed growth

Let ξ \xi be a real irrational number. We are interested in sequences of linear forms in 1 and ξ \xi , with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of ξ \xi . We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin’s new proof of Nesterenko’s linear independence criterion.

On the expansion of some exponential periods in an integer base

We derive a lower bound for the subword complexity of the base-b expansion (b ≥ 2) of all real numbers whose irrationality exponent is equal to 2. This provides a generalization of a theorem due to Ferenczi and Mauduit. As a consequence, we obtain the first lower bound for the subword complexity of the number e and of some other transcendental exponential periods.

Irrationality measures for some automatic real numbers

AbstractThis paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue–Morse, Rudin–Shapiro, paperfolding and Baum–Sweet sequences. These upper bounds arise from the construction of some explicit Padé or Padé type approximants for the generating functions of these sequences. In particular, we prove that the Thue–Morse–Mahler numbers have an irrationality exponent at most equal to 4. We also obtain an explicit description of infinitely many convergents to these numbers.

Impossibility of extending Pólya’s theorem to “forms” with arbitrary real exponents

Pólya proved that if a form (homogeneous polynomial) with real coefficients is positive on the nonnegative orthant (except at the origin), then it is the quotient of two real forms with no negative coefficients. While Pólya’s theorem extends, easily, from ordinary real forms to “generalized” real forms with arbitrary rational exponents, we show that it does not extend to generalized real forms with arbitrary real (possibly irrational) exponents.