Some generalizations of Schmidt's subspace theorem
Si Duc Quang
备注：
In this version, some typos are corrected and the references list is modified. The paper consists of 17 pages. arXiv admin note: substantial text overlap with arXiv:1808.10286; text overlap with arXiv:math/0408381, arXiv:1910.07966 by other authors
The aim of this paper is twofold. The first is to give a quantitative version of Schmidt's subspace theorem for arbitrary families of higher degree polynomials. The second is to give a generalization of the subspace theorem for arbitrary families of closed subschemes in algebraic projective varieties.
We deduce an effective version of Schmidt's subspace theorem on a smooth projective variety X over function fields of characteristic zero for hypersurfaces located in N-subgeneral position with respect to X.
We study extensions and generalizations of the Schmidt Subspace Theorem in various settings. In particular, we prove results for algebraic points of bounded degree, giving a sharp version of Schmidt's theorem for quadratic points in the projective plane and a more general result that resolves a conjecture of Schlickewei.
In this paper, we establish a Schmidt's subspace theorem for moving hypersurfaces in weakly subgeneral position. Our result generalizes the previous results on Schmidt's theorem for the case of moving hypersurfaces.
We establish an effective version of Schmidt's subspace theorem on a smooth projective variety $\mathcal{X}$ over function fields of characteristic zero for hypersurfaces located in m-subgeneral position with respect to $\mathcal{X}$. Our result generalizes and improves the results of An-Wang, Ru-Wang and the author.
Recently, Xie-Cao [15] obtained a Second Main Theorem for moving hypersurfaces located in subgeneral position with index which is extended the result of Ru [11]. By using some methods due to Son-Tan-Thin [13], Quang [9] and Xie-Cao [15], we shall give a Schmidt's Subspace Theorem for moving hypersurface targets in subgeneral position with index intersecting algebraic variety. Our result is a extension the Schmidt's Subspace Theorem due to Son-Tan-Thin [13] and Quang [9].
It was discovered that there is a formal analogy between Nevanlinna theory and Diophantine approximation. Via Vojta's dictionary, the Second Main Theorem in Nevanlinna theory corresponds to Schmidt's Subspace Theorem in Diophantine approximation. Recently, Cherry, Dethloff, and Tan (arXiv:1503.08801v2 [math.CV]) obtained a Second Main Theorem for moving hypersurfaces intersecting projective varieites. In this paper, we shall give the counterpart of their Second Main Theorem in Diophantine approximation.
In this paper, we prove a generalization of the Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety over a number field. Our result improves and generalizes the previous results on Schmidt's subspace theorem for the case of higher degree polynomials.
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.
We prove a generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of suitably defined Seshadri constants with respect to a fixed ample divisor. Our proof builds on previous work by Evertse and Ferretti, Corvaja and Zannier, and others, and uses standard techniques from algebraic geometry such as notions of positivity, blowing-ups and direct image sheaves. As an application, we recover a higher-dimensional Diophantine approximation theorem of K.F. Roth-type due to D. McKinnon and M. Roth with a significantly shortened proof, while simultaneously ext…
This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem…
We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $α_1x_1+α_2x_2 +y$ with {\it positive} integers $x_1,x_2$.
In 1967, Schmidt wrote a seminal paper [10] on heights of subspaces of R n or C n defined over a number field K, and diophantine approximation problems. The going-down Theorem -- one of the main theorems he proved in his paper -- remains valid in two cases depending on whether the embedding of K in the complex field C is a real or a complex non-real embedding. For the latter, and more generally as soon as K is not totally real, at some point of the proof, the arguments in [10] do not exactly work as announced. In this note, Schmidt's ideas are worked out in details and his proof of the complex…
This work is motivated by a paper of Davenport and Schmidt, which treats the question of when Dirichlet's theorems on the rational approximation of one or of two irrationals can be improved and if so, by how much. We consider a generalization of this question in the simplest case of a single irrational but in the context of the geometry of numbers in $\mathbb R^2$, with the sup-norm replaced by a more general one. Results include sharp bounds for how much improvement is possible under various conditions. The proofs use semi-regular continued fractions that are characterized by a certain best a…
In this paper, we extend the recent theorem of G. Heier and A. Levin [arXiv:1712.02456] on the generalization of Schmidt's subspace theorem and Cartan's Second Main Theorem in Nevanlinna theory to closed subschemes located in $l$-subgeneral position, using the generic linear combination technique due to Quang.
We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasi-periodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to any real algebraic number.
The Schmidt Subspace Theorem affirms that the solutions of some particular system of diophantine approximations in projective spaces accumulates on a finite number of proper linear subspaces. Given a subvariety $X$ of a projective space $P^n$, does there exists a system of diophantine approximations on $P^n$ whose solutions are Zariski dense in $P^n$, but lie in finitely many proper subvarieties of $X$? One can gain insight into this problem using a theorem of G. Faltings and G. Wüstholz. Their construction requires the hypothesis that the sum of some expected values has to be large. This sum …
We use a generalization of Vinogradov's mean value theorem of S. Parsell, S. Prendiville and T. Wooley and ideas of W. Schmidt to give nontrivial bounds for the number of solutions to polynomial congruences, for arbitrary polynomials, when the solutions lie in a very general class of sets, including all convex sets.
Let $S=(s_n)_{n\geq 1}$ be a sequence with elements in a commutative monoid $(\mathcal{M},+,0)$. In this paper, we provide an explicit formula for $$\sum_{\la} C(\la) q^{\sum_{n\geq 1} \la_n\cdot s_n}$$ where $\la=(\la_1,\ldots)$ run through some subsets of over-partitions, and $C(\la)$ is a certain product of ``colors'' assigned to the parts of $\la$, and $q^s$ is a formal power of $q$ for $s\in M$. This formula allows us not only to retrieve several known Schmidt-type theorems but also to provide new Schmidt-type theorems for non-periodic sequences $S$. For example, when $(M,+,0)=(\mathbb{Z}…
