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1 
Tools and techniques in diophantine approximationHaynes, Alan Kaan, January 1900 (has links) (PDF)
Thesis (Ph. D.)University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

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Tools and techniques in diophantine approximationHaynes, Alan Kaan 28 August 2008 (has links)
Not available / text

3 
On some distribution problems in Analytic Number TheoryHomma, Kosuke 26 August 2010 (has links)
This dissertation consists of three parts. In the first part we consider the equidistribution of roots of quadratic congruences. The roots of quadratic congruences are known to be equidistributed. However,we establish a bound for the discrepancy of this sequence using a spectral method involvingautomorphic forms, especially Kuznetsov's formula, together with an ErdősTurán inequality. Then we discuss the implications of our discrepancy estimate for the reducibility problem of arctangents of integers. In the second and third part of this dissertation we consider some aspects of Farey fractions. The set of Farey fractions of order at most [mathematical formula] is, of course, a classical object in Analytic Number Theory. Our interest here is in certain sumsets of Farey fractions. Also, in this dissertation we study Farey fractions by working in the quotient group Q/Z, which is the modern point of view. We first derive an identity which involves the structure of Farey fractions in the group ring of Q/Z. Then we use these identities to estimate the asymptotic magnitude of the size of the sumset [mathematical formula]. Our method uses results about divisors in short intervals due to K. Ford. We also prove a new form of the ErdősTurán inequality in which the usual complex exponential functions are replaced by a special family of functions which are orthogonal in L²(R/Z). / text

4 
Fourier expansions for Eisenstein series twisted by modular symbols and the distribution of multiples of real points on an elliptic curveCowan, Alexander January 2019 (has links)
This thesis consists of two unrelated parts.
In the first part of this thesis, we give explicit expressions for the Fourier coefficients of Eisenstein series E∗(z, s, χ) twisted by modular symbols ⟨γ, f⟩ in the case where the level of f is prime and equal to the conductor of the Dirichlet character χ. We obtain these expressions by computing the spectral decomposition of an automorphic function closely related to E∗(z, s, χ). We then give applications of these expressions. In particular, we evaluate sums such as Σχ(γ)⟨γ, f⟩, where the sum is over γ ∈ Γ∞\Γ0(N) with c^2 + d^2 < X, with c and d being the lowerleft and lowerright entries of γ respectively. This parallels past work of Goldfeld, Petridis, and Risager, and we observe that these sums exhibit different amounts of cancellation than what one might expect.
In the second part of this thesis, given an elliptic curve E and a point P in E(R), we investigate the distribution of the points nP as n varies over the integers, giving bounds on the x and y coordinates of nP and determining the natural density of integers n for which nP lies in an arbitrary open subset of {R}^2. Our proofs rely on a connection to classical topics in the theory of Diophantine approximation.

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The hypoellipticity of differential forms on closed manifoldsWenyi, Chen, Tianbo, Wang January 2005 (has links)
In this paper we consider the hypoellipticity of differential forms on a closed manifold.The main results show that there are some topological obstruct for the existence of the differential forms with hypoellipticity.

6 
Questions Involving Countable Intersection GamesAtchley, James Holmes 07 1900 (has links)
We consider questions involving two different variations of Schmidt's game: the rho game and the HAW (Hyperplane Absolutely Winning) game.

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Reduced Ideals and Periodic Sequences in Pure Cubic FieldsJacobs, G. Tony 08 1900 (has links)
The “infrastructure” of quadratic fields is a body of theory developed by Dan Shanks, Richard Mollin and others, in which they relate “reduced ideals” in the rings and subrings of integers in quadratic fields with periodicity in continued fraction expansions of quadratic numbers. In this thesis, we develop cubic analogs for several infrastructure theorems. We work in the field K=Q(), where 3=m for some squarefree integer m, not congruent to ±1, modulo 9. First, we generalize the definition of a reduced ideal so that it applies to K, or to any number field. Then we show that K has only finitely many reduced ideals, and provide an algorithm for listing them. Next, we define a sequence based on the number alpha that is periodic and corresponds to the finite set of reduced principal ideals in K. Using this rudimentary infrastructure, we are able to establish results about fundamental units and reduced ideals for some classes of pure cubic fields. We also introduce an application to Diophantine approximation, in which we present a 2dimensional analog of the Lagrange value of a badly approximable number, and calculate some examples.

8 
Parametric Geometry of NumbersRivardCooke, Martin 06 March 2019 (has links)
This thesis is primarily concerned in studying the relationship between different exponents of Diophantine approximation, which are quantities arising naturally in the study of rational approximation to a fixed ntuple of real irrational numbers.
As Khinchin observed, these exponents are not independent of each other, spurring interest in the study of the spectrum of a given family of exponents, which is the set of all possible values that can be taken by said family of exponents.
Introduced in 20092013 by Schmidt and Summerer and completed by Roy in 2015, the parametric geometry of numbers provides strong tools with regards to the study of exponents of Diophantine approximation and their associated spectra by the introduction of combinatorial objects called nsystems. Roy proved the very surprising result that the study of spectra of exponents is equivalent to the study of certain quantities attached to nsystems. Thus, the study of rational approximation can be replaced by the study of nsystems when attempting to determine such spectra.
Recently, Roy proved two new results for the case n=3, the first being that spectra are semialgebraic sets, and the second being that spectra are stable under the minimum with respect to the product ordering. In this thesis, it is shown that both of these results do not hold in general for n>3, and examples are given.
This thesis also provides nontrivial examples for n=4 where the spectra is stable under the minimum.
An alternate and much simpler proof of a recent result of MarnatMoshchevitin proving an important conjecture of SchmidtSummerer is also given, relying only on the parametric geometry of numbers instead. Further, a conjecture which generalizes this result is also established, and some partial results are given towards its validity. Among these results, the simplest, but nontrivial, new case is also proven to be true.
In a different vein, this thesis considers certain generalizations theta(q) of the classical theta qseries. We show under conditions on the coefficients of the series that theta(q) is neither rational nor quadratic irrational for each integer q>1.

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Constructing Simultaneous Diophantine Approximations Of Certain Cubic NumbersHinkel, Dustin January 2014 (has links)
For K a cubic field with only one real embedding and α, β ϵ K, we show how to construct an increasing sequence {m_n} of positive integers and a subsequence {ψ_n} such that (for some constructible constants γ₁, γ₂ > 0): max{ǁm_nαǁ,ǁm_nβǁ} < [(γ₁)/(m_n^(¹/²))] and ǁψ_nαǁ < γ₂/[ψ_n^(¹/²) log ψ_n] for all n. As a consequence, we have ψ_nǁψ_nαǁǁψ_nβǁ < [(γ₁ γ₂)/(log ψ_n)] for all n, thus giving an effective proof of Littlewood's conjecture for the pair (α, β). Our proofs are elementary and use only standard results from algebraic number theory and the theory of continued fractions.

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Metric number theory : the good and the badThorn, Rebecca Emily January 2005 (has links)
Each aspect of this thesis is motivated by the recent paper of Beresnevich, Dickinson and Velani (BDV03]. Let 'ljJ be a real, positive, decreasing function i.e. an approximation function. Their paper considers a general lim sup set A( 'ljJ), within a compact metric measure space (0, d, m), consisting of points that sit in infinitely many balls each centred at an element ROt of a countable set and of radius 'I/J(130) where 130 is a 'weight' assigned to each ROt. The classical set of 'I/Jwell approximable numbers is the basic example. For the set A('ljJ) , [BDV03] achieves mmeasure and Hausdorff measure laws analogous to the classical theorems of Khintchine and Jarnik. Our first results obtain an application of these metric laws to the set of 'ljJwell approximable numbers with restricted rationals, previously considered by Harman (Har88c]. Next, we consider a generalisation of the set of badly approximable numbers, Bad. For an approximation function p, a point x of a compact metric space is in a general set Bad(p) if, loosely speaking, x 'avoids' any ball centred at an element ROt of a countable set and of radius c p(I3Ot) for c = c(x) a constant. In view of Jarnik's 1928 result that dim Bad = 1, we aim to show the general set Bad(p) has maximal Hausdorff dimension. Finally, we extend the theory of (BDV03] by constructing a general lim sup set dependent on two approximation functions, A('ljJll'ljJ2)' We state a measure theorem for this set analogous to Khintchine's (1926a) theorem for the Lebesgue measure of the set of ('l/Jl, 1/12)well approximable pairs in R2. We also remark on the set's Hausdorff dimension.

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