Subspace Theorem Research Articles

On the transcendance of quasi-periodic Rosen continued fractions

AbstractIn this paper, we consider the two Hecke groups $$G_{4}$$ G 4 and $$G_{6}$$ G 6 and we use the Schmidt Subspace Theorem to establish the transcendence of some quasi-periodic Rosen continued fractions in order to get the exact analogues of the results established with the regular continued fractions.

GCD inequalities inspired by Vojta’s conjecture

AbstractWe prove several GCD inequalities inspired by Vojta’s deep conjecture in Diophantine geometry. These inequalities work with all algebraic numbers, so they can be thought of as attempts to extend GCD inequalities for S-units obtained by Corvaja and Zannier. Some of the inequalities we derive are weakened versions of inequalities obtained by Silverman under the assumption of Vojta’s conjecture. The upper bounds for the GCD’s in these inequalities all involve some contributions of heights and some contributions coming from the local heights outside S. The main ingredient of the proofs is the recent Diophantine approximation result of Ru and Vojta which is based on Schmidt subspace theorem and which involves a birational invariant coming from the asymptotic growth of the number of global sections of certain line bundles.

Transcendence of certain sequences of algebraic numbers

Using Schmidt’s Subspace Theorem, this paper improves and extends an existing transcendence result for sequences of algebraic numbers. The theorems thus produced correspond to a central theorem on the irrationality of sequences due to Erdős.

On non-Zariski density of (D,S)-integral points in forward orbits and the Subspace Theorem

Working over a base number field K, we study the attractive question of Zariski non-density for (D,S)-integral points in Of(x) the forward f-orbit of a rational point x∈X(K). Here, f:X→X is a regular surjective self-map for X a geometrically irreducible projective variety over K. Given a non-zero and effective f-quasi-polarizable Cartier divisor D on X and defined over K, our main result gives a sufficient condition, that is formulated in terms of the f-dynamics of D, for non-Zariski density of certain dynamically defined subsets of Of(x). For the case of (D,S)-integral points, this result gives a sufficient condition for non-Zariski density of integral points in Of(x). Our approach expands on that of Yasufuku, [13], building on earlier work of Silverman [11]. Our main result gives an unconditional form of the main results of [13]; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in [10] and expanded upon in [3] and [6].

On Absolute and Quantitative Subspace Theorems

Abstract The Absolute Subspace Theorem, a vast generalization and a quantitative improvement of Schmidt’s Subspace Theorem, was first established by Evertse and Schlickewei and then strengthened remarkably by Evertse and Ferretti. We study quantitative generalizations and extensions of subspace theorems in various contexts. We establish a generalization of Evertse and Ferretti’s Absolute Subspace Theorem for hyperplanes in general position. We obtain improved (non-absolute) Quantitative Subspace Theorems for hyperplanes in general position and in subgeneral position. We show a Semi-quantitative Subspace Theorem for hyperplanes in non-subdegenerate position.

Analytic representations of backward shifts on weighted Bergman and Dirichlet spaces

The backward shifts (the adjoints of multiplication by z) on weighted Bergman spaces of the unit disk are important in operator theory. For example, Agler [1] shows their restrictions to invariant subspaces of vector-valued weighted Bergman spaces are model operators for hypercontractions. We present analytic representations of backward shifts on weighted Bergman spaces and Dirichlet type spaces. Since these analytic representations are not space specific, it is interesting to study these operators on Hilbert spaces of holomorprhic functions on the unit disk. We initiate this study by developing several properties of these operators. In particular, we obtain simple proofs and extend the invariant subspace theorem of shift plus Volterra operator on the Hardy space by Čučković and Paudyal [16]. Furthermore, we describe invariant subspaces of shift plus Volterra operator on the Bergman space by relating them to the invariant subspaces of the Dirichlet shift. Analytic representations of the tuple of backward shifts on weighted Bergman spaces of the unit ball are also discussed.

Schmidt's subspace theorem for arbitrary families of moving hypersurface targets

In this article, we mainly obtain a Schmidt's Subspace Theorem for arbitrary families of moving hypersurface targets in algebraic varieties, which is an extension of the Schmidt's Subspace Theorem due to Son-Tan-Thin [17] and Quang [10], etc.

A subspace theorem for manifolds

We prove a theorem that generalizes Schmidt’s Subspace Theorem in the context of metric diophantine approximation. To do so we reformulate the Subspace Theorem in the framework of homogeneous dynamics by introducing and studying a slope formalism and the corresponding notion of semistability for diagonal flows.

Diophantische Approximationen

This workshop was focused on a large variety of problems which have seen important progress during the last few years, such as extensions and refinements of Schmidt Subspace Theorem, together with new applications, works on the Zilber-Pink conjecture and on unlikely intersections, geometry of numbers, simultaneous Diophantine approximation, theory of heights, continued fractions, and arithmetic dynamics.

Cyclic multiplicity of a direct sum of forward and backward shifts

Let E E and F F be two complex Hilbert spaces. Let S E S_{E} and S F S_{F} be the shift operators on vector-valued Hardy space H E 2 H_{E}^{2} and H F 2 H_{F}^{2} , respectively. We show that the cyclic multiplicity of S E ⊕ S F ∗ S_{E}\oplus S_{F} ^{\ast } equals 1 + dim ⁡ E 1+\dim E . This result is classical when dim ⁡ E = dim ⁡ F = 1 \dim E=\dim F=1 (see J. A. Deddens [On A ⊕ A ∗ A \oplus A^\ast , 1972]; Paul Richard Halmos [A Hilbert space problem book, Springer-Verlag, New York-Berlin, 1982]; Domingo A. Herrero and Warren R. Wogen [Rocky Mountain J. Math. 20 (1990), pp. 445–466]). Our approach is inspired by the elegant and short proof of this classical result attributed to Nikolskii, Peller and Vasunin in Halmos’s book [A Hilbert space problem book, Springer-Verlag, New York-Berlin, 1982]. By using the invariant subspace theorems for S E ⊕ S F ∗ S_{E}\oplus S_{F}^{\ast } (see M. C. Câmara and W. T. Ross [Canad. Math. Bull. 64 (2021), pp. 98–111]; Caixing Gu and Shuaibing Luo [J. Funct. Anal. 282 (2022), 31 pp.]; Dan Timotin [Concr. Oper. 7 (2020), pp. 116–123]), we characterize non-cyclic subspaces of S E ⊕ S F ∗ S_{E}\oplus S_{F}^{\ast } when dim ⁡ E > ∞ \dim E>\infty and dim ⁡ F = 1 \dim F=1 .

Regular covariant representations and their Wold-type decomposition

Olofsson introduced a growth condition regarding elements of an orbit for an expansive operator and generalized Richter's wandering subspace theorem. Later on, using the Moore-Penrose inverse, Ezzahraoui, Mbekhta, and Zerouali extended the growth condition and obtained a Shimorin-Wold-type decomposition. Shimorin-Wold-type decomposition for completely bounded covariant representations, which are close to isometric representations, is obtained in [25]. This paper extends this decomposition for regular, completely bounded covariant representation having reduced minimum modulus ≥1 that satisfies the growth condition. To prove the decomposition, we introduce the terms regular, algebraic core, and reduced minimum modulus in the completely bounded covariant representation setting and work out several fundamental results. Consequently, we shall analyze the weighted unilateral shift introduced by Muhly and Solel and introduce and explore a non-commutative weighted bilateral shift.

Transcendental series of reciprocals of Fibonacci and Lucas numbers

Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erdos and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is irrational for any sequence of positive integers $n_1,n_2,\ldots$ with $\frac{n_{k+1}}{n_k}\geq c>1$. We resolve the transcendence counterpart of their question: as a special case of our main theorem, we have that $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is transcendental when $\frac{n_{k+1}}{n_k}\geq c>2$. The bound $c>2$ is best possible thanks to the identity at the beginning. This paper provides a new way to apply the Subspace Theorem to obtain transcendence results and extends previous non-trivial results obtainable by only Mahler's method for special sequences of the form $n_k=d^k+r$.

On simultaneous approximation of algebraic numbers

Let $\Gamma\subset \bar{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $\alpha_1,\ldots,\alpha_r\in\bar{\mathbb Q}^\times$ be algebraic numbers which are $\mathbb{Q}$-linearly independent and let $\epsilon>0$ be a given real number. One of the main results that we prove in this article is as follows; There exist only finitely many tuples $(u, q, p_1,\ldots,p_r)\in\Gamma\times\mathbb{Z}^{r+1}$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ for some integer $d\geq 1$ satisfying $|\alpha_i q u|>1$, $\alpha_i q u$ is not a pseudo-Pisot number for some integer $i\in\{1, \ldots, r\}$ and $$ 0<|\alpha_j qu-p_j|<\frac{1}{H^\epsilon(u)|q|^{\frac{d}{r}+\varepsilon}} $$ for all integers $j = 1, 2,\ldots, r$, where $H(u)$ is the absolute Weil height. In particular, when $r =1$, this result was proved by Corvaja and Zannier in [3]. As an application of our result, we also prove a transcendence criterion which generalizes a result of Han\v{c}l, Kolouch, Pulcerov\'a and \v{S}t\v{e}pni\v{c}ka in [4]. The proofs rely on the clever use of the subspace theorem and the underlying ideas from the work of Corvaja and Zannier.

An effective Schmidt’s subspace theorem for arbitrary hypersurfaces over function fields

We establish an effective version of Schmidt’s subspace theorem on a smooth projective variety [Formula: see text] over function fields of characteristic zero for arbitrary family of hypersurfaces. We also show an application to study a general Thue equation over function fields.

Generalizations of degeneracy second main theorem and Schmidt’s subspace theorem

In this paper, by introducing the notion of "\textit{distributive constant}" of a family of hypersurfaces with respect to a projective variety, we prove a second main theorem in Nevanlinna theory for meromorphic mappings with arbitrary families of hypersurfaces in projective varieties. Our second main theorem generalizes and improves previous results for meromorphic mappings with hypersurfaces, in particular for algebraically degenerate mappings and for the families of hypersurfaces in subgeneral position. The analogous results for the holomorphic curves with finite growth index from a complex disc into a project variety, and for meromorphic mappings on a complete K\"{a}hler manifold are also given. For the last aim, we will prove a Schmidt's subspace theorem for an arbitrary families of homogeneous polynomials, which is the counterpart in Number theory of our second main theorem. Our these results are generalizations and improvements of all previous results.

Algebraic Number Starscapes

We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by the resulting images, which we have called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focusing on the quadratic and cubic cases. The geometry describes and explains the notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. Meanwhile, the images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. In particular, the paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural action. Hyperbolic geometry and the discriminant play an important role in low degree. In particular, we rediscover the quadratic and cubic root formulas as isometries of and its unit tangent bundle, respectively. Utilizing this geometry, we determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We reconsider the fundamental questions of the Diophantine approximation of complex numbers by algebraic numbers of bounded degree, from the geometric perspective developed. In the quadratic case (approximation by quadratic irrationals), we consider approximation in terms of hyperbolic distance between roots in the complex plane and the discriminant as a measure of arithmetic height on a polynomial. In particular, we determine the supremum on the exponent k for which an algebraic target α has infinitely many approximations β whose hyperbolic distance from α does not exceed . It turns out to fall into two cases, depending on whether α lies on the image of a plane of rational slope in coefficient space (a rational geodesic). The result comes as an application of Schmidt’s subspace theorem. Our results recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered. Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.

An effective function field version of Schmidt's subspace theorem for projective varieties, with arbitrary families of homogeneous polynomials

In this paper, by giving a new lower bound estimate for the Chow weight of a projective variety, we will prove an effective Schmidt subspace theorem for projective varieties on the rational function field with arbitrary families of higher degree homogeneous polynomials. Our result is a generalization and also an improvement of the previous results.

Conditions for matchability in groups and field extensions

The origins of the notion of matchings in groups spawn from a linear algebra problem proposed by E. K. Wakeford [On canonical forms. Proc London Math Soc (2). 1920;18:403–410] which was tackled in Fan and Losonczy [Matchings and canonical forms for symmetric tensors. Adv Math. 1996;117(2):228–238]. In this paper, we first discuss unmatchable subsets in abelian groups. Then we formulate and prove linear analogues of results concerning matchings, along with a conjecture that, if true, would extend the primitive subspace theorem. We discuss the dimension m-intersection property for vector spaces and its connection to matching subspaces in a field extension, and we prove the linear version of an intersection property result of certain subsets of a given set.

A quantitative subspace Balian-Low theorem

Let G⊂L2(R) be the subspace spanned by a Gabor Riesz sequence (g,Λ) with g∈L2(R) and a lattice Λ⊂R2 of rational density. It was shown recently that if g is well-localized both in time and frequency, then G cannot contain any time-frequency shift π(z)g of g with z∈R2∖Λ. In this paper, we improve the result to the quantitative statement that the L2-distance of π(z)g to the space G is equivalent to the Euclidean distance of z to the lattice Λ, in the sense that the ratio between those two distances is uniformly bounded above and below by positive constants. On the way, we prove several results of independent interest, one of them being closely related to the so-called weak Balian-Low theorem for subspaces.

Diophantine approximation

This article gives an introductory survey of recent progress on Diophantine problems, especially consequences coming from Schmidt’s subspace theorem, Baker’s transcendence method and Padé approximation. We present fundamental properties around Diophantine approximation and how it yields results in number theory.