Rational and algebraic approximations of algebraic numbers and their application

Previous article Next article Numerical Evaluation of Continued FractionsG. BlanchG. Blanchhttps://doi.org/10.1137/1006092PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Leo A. Aroian, Continued fractions for the incomplete Beta function, Ann. Math. Statistics, 12 (1941), 218–223 MR0005193 0025.31901 CrossrefGoogle Scholar[2] Friedrich L. Bauer, The quotient-difference and epsilon algorithms, Edited by R. E. Langer. Publication no. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin, The University of Wisconsin Press, Madison, 1959x+462 MR0102594 Google Scholar[3] Gertrude Blanch, On the computation of Mathieu functions, J. Math. Phys. Mass. Inst. Tech., 25 (1946), 1–20 MR0016690 0061.27702 ISIGoogle Scholar[4] Gertrude Blanch and , Ida Rhodes, Table of characteristic values of Mathieu's equation for large values of the parameter, J. Washington Acad. 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Let Q denote the rational number field and α be an algebraic number of degree s. Then the algebraic number field F s = Q(α) is the field given by the polynomials in α of degree < s with rational coefficients.

Diophantische Approximationen

Diophantine approximation is concerned with the quantitative study of the density of the rational numbers inside the real numbers. The Diophantine properties of a real number can be quantified through its approximation properties by rational (and more generally algebraic) numbers. For rational approximation, continued fractions provide an important tool in studying such properties. For higher dimensional problems and for algebraic approximation, different methods are needed. The metric theory of Diophantine approximation is concerned with the size of sets of numbers enjoying specified Diophantine properties. It is a general feature of the theory that most natural properties give rise to zero–one laws: the set of numbers enjoying the property in question is either null or full with respect to the Lebesgue measure. A more refined study of the null sets can be done using the notions of Hausdorff measure and dimension. Over the years, considerable work has gone into studying metric Diophantine approximation on subsets of $$\mathbb {R}^n$$ . The initial focus was on curves, surfaces and manifolds, but in recent years much effort has also gone into the study of fractal subsets. Already in the setting of rational approximation of real numbers, many problems which seem simple enough remain open. For instance, it is not known whether the Cantor middle third set contains an algebraic, irrational number (it is conjectured not to do so). In these notes, starting from the classical setup, I will work towards the study of metric Diophantine approximation on fractal sets. Along the way, we will touch upon some major open problems in Diophantine approximation, such as the Littlewood conjecture and the Duffin–Schaeffer conjecture; and newer methods originating in ergodic theory and dynamical systems will also be discussed. The required elements from fractal geometry will be covered.

Qualitative properties of solutions for an integral system related to the Hardy–Sobolev inequality

The Siegel conjecture on the rational approximation to algebraic numbers was proved a few years ago by K. F. Roth [1] with the following theorem: Let α be any algebraic number, not rational. If has an infinity of solutions in integers h and q (q &gt; 0) tehn k ≤ 2.

The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications. As for the reason that in most cases the offset curves do not preserve the same polynomial or rational polynomial representations, it arouses difficulty in applications. Thus approximation methods have been introduced to solve this problem. In this paper, it has been pointed out that the crux of offset curve approximation lies in the approximation of parametric speed. Based on the Jacobi polynomial approximation theory with endpoints interpolation, an algebraic rational approximation algorithm of offset curve, which preserves the direction of normal, is presented.

We study rational approximations x / y x/y to algebraic and, more generally, to real numbers ξ \xi . Given δ &gt; 0 \delta &gt; 0 , and writing L = log ⁡ ( 1 + δ ) L = \log (1 + \delta ) , the number of approximations with | ξ − ( x / y ) | &gt; y − 2 − δ |\xi - (x/y)| &gt; {y^{ - 2 - \delta }} is ≤ L − 1 log ⁡ log ⁡ H + c 1 ( δ , r ) \leq {L^{ - 1}}\log \log H + {c_1}(\delta ,r) if ξ \xi is algebraic of degree ≤ r \leq r and of height H H , and is ≤ L − 1 log ⁡ log ⁡ B + c 2 ( δ ) \leq {L^{ - 1}}\log \log B + {c_2}(\delta ) if ξ \xi is real and we restrict to approximations with y ≤ B y \leq B . It turns out that the dependency on H H resp. B B in these estimates is the best possible, i.e., that the summands L − 1 log ⁡ log ⁡ H {L^{ - 1}}\log \log H resp. L − 1 log ⁡ log ⁡ B {L^{ - 1}}\log \log B are optimal.

We study rational approximations x/y to algebraic and, more generally, to real numbers { .Given > 0 , and writing L = log(l + c5)^the number of approximations with | -(x/y)\ < y~2~s is < L_1 loglog// + C\{S,r) if ( is algebraic of degree < r and of height H , and is < L_l loglog#+C2(<5) if is real and we restrict to approximations with y < B .It turns out that the dependency on H resp. B in these estimates is the best possible, i.e., that the summands Z.-1 loglog// resp.L~'loglog are optimal.

Let F s = Q(α) be a real algebraic number field of degree s. We shall give in this chapter an algorithm for the simultaneous Diophantine approximation obtained by η l = α l (l = 1, 2, ....) which is essentially the Jacobi-Perron algorithm (Cf. L. Bernstein [1]). It yields less precise results but the computations of nl and h lj .(1 ≤ j ≤ s) are comparatively simple.

The Hexic transform ρ of the noncommutative 2-torus Aθ is the canonical order 6 automorphism defined by ρ(U)=V, ρ(V)=e−πiθU−1V, where U, V are the canonical unitary generators obeying the unitary Heisenberg commutation relation VU=e2πiθUV. The Cubic transform is κ=ρ2. These are canonical analogues of the noncommutative Fourier transform, and their associated fixed point C⁎-algebras Aθρ, Aθκ are noncommutative Z6, Z3 toroidal orbifolds, respectively. For a large class of irrationals θ and rational approximations p/q of θ, a projection e of trace q2θ−pq is constructed in Aθ that is invariant under the Hexic transform. Further, this projection is shown to be a matrix projection in the sense that it is approximately central, the cut down algebra eAθe contains a Hexic invariant q×q matrix algebra M whose unit is e and such that the cut downs eUe, eVe are approximately inside M. It is also shown that these invariant matrix projections are covariant in that they arise from a continuous section E(t) of C∞-projections of the continuous field {At}0<t<1 of noncommutative tori C⁎-algebras such that ρ(E(t))=E(t). It turns out that the projection E(t) is the support projection of a canonical C∞-positive element that has the appearance of a noncommutative 2-dimensional Theta function. The topological invariants (or ‘quantum’ numbers) of E(t), e, and related projections are computed by a new and quicker method than in previous works. (They would also give topological invariants for finitely generated projective modules over noncommutative orbifolds associated to Z6 and Z3 symmetries of noncommutative tori.) We remark that these results have some bearing on research work related to noncommutative orbifolds used in string theory.

In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the sequence of classical Fourier coefficients by a short exponential sum with a pre-defined number of $$N+1$$ terms. The obtained approximation can be interpreted as an adaptive N-th Fourier sum with respect to the orthogonal Takenaka-Malmquist basis. Using the theoretical results on rational approximation in Hardy spaces and on the decay of singular values of special infinite Hankel matrices, we show that adaptive Fourier sums can converge essentially faster than classical Fourier sums for a large class of functions. Further, we derive an algorithm to compute almost optimal adaptive Fourier sums. Our numerical results show that the significantly better convergence behavior of adaptive Fourier sums for optimally chosen basis elements can also be achieved in practice. For comparison, we also provide a greedy algorithm to determine an adaptive Fourier sum. This algorithm requires less computational effort but yields essentially slower convergence.

When encoding real numbers as (necessarily infinite) bit-strings, the naïve binary/decimal expansion is well-known [ doi:10.1112/plms/s2-43.6.544 ] computably “ un reasonable”, rendering, for example, tripling qualitatively discontinuous on Cantor’s sequence space. Encoding reals as sequences of (finite integer numerators and denominators, in binary, of) rational approximations does make common operations qualitatively computable, yet admits no bounds on their computational complexity/quantitative continuity. Dyadic approximations, on the other hand, are known polynomially, and signed binary expansions even linearly, “reasonable” in a rigorous sense recalled in the introduction of this work. But how to distinguish between un/suitable encodings of spaces common in Calculus beyond the reals, such as Banach or Sobolev? With respect to qualitative computability/continuity on topological spaces, the technical condition of admissibility had been identified [ doi:10.1016/0304-3975(85)90208-7 ] for an encoding over Cantor space (historically called a representation ) to be “reasonable” [ doi:10.1007/978-3-030-59234-9_9 ] . Roughly speaking, admissibility requires the representation to be (i) continuous, and to be (ii) maximal with respect to continuous reduction. Admissible representations exist for a large class of spaces. And for (precisely) these does the Kreitz–Weihrauch—sometimes aka Main —Theorem of Computable Analysis hold, which characterizes continuity of functions by continuity of mappings translating codes, so-called realizers . We refine qualitative computability/continuity on topological spaces to quantitative continuity/complexity on metric spaces by proposing a notion, and investigating the properties, of polynomially/linearly admissible representations. Roughly speaking, these are (i) close to “optimally” continuous, namely linearly/polynomially relative to the space’s entropy, and they are (ii) maximal with respect to relative linear/polynomial quantitatively continuous reductions defined in the main text. Quantitatively admissible representations are closed under composition over generalized ground spaces beyond Cantor’s. Such representations exhibit a quantitative strengthening of the qualitative Main Theorem , namely now characterizing quantitative continuity of functions by quantitative continuity of realizers. A large class of compact metric spaces is shown to admit polynomially admissible representations over compact ultra metric spaces, and some even a generalization of the linearly admissible signed binary encoding. Quantitative admissibility thus provides the desired criterion for complexity-theoretically “reasonable” encodings.

In this paper, using only the St$ \ddot{o} $rmer theorem and its generalizations on Pell's equation and fundamental properties of Lehmer sequence and the associated Lehmer sequence, we discuss the Diophantine equations $x^2-Dy^2 = -1$ and $x^2-Dy^2 = 4$. We obtain the relation between a positive integer solution (x, y) of the Diophantine equation $x^2-Dy^2 = -1$ and its fundamental solution if there is exactly one or two prime divisors of y not dividing D. We also obtain the relation between a positive integer solution (x, y) of the Diophantine equation $x^2-Dy^2 = 4$ and its minimal positive solution if there is exactly two prime divisors of y not dividing D.

Klaus Friedrich Roth, 1925-2015