Subspace Theorem Research Articles

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.

Schmidt’s Subspace Theorem for Moving Hypersurface Targets in Subgeneral Position in Projective Varieties

Schmidt’s Subspace Theorem for Moving Hypersurface Targets in Subgeneral Position in Projective Varieties

Quantitative subspace theorem and general form of second main theorem for higher degree polynomials

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 in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions.

Schmidt’s subspace theorem for non-subdegenerate families of hyperplanes

In this paper, we establish a Schmidt’s subspace theorem for non-subdegenerate families of hyperplanes. In particular, our result improves the previous result on Schmidt’s subspace type theorem for the case of non-degenerate families of hyperplanes, and furthermore, also shows the sharpness of the condition of non-subdegeneracy. As a consequence, we deduce a version of Lang’s conjecture on exceptional sets in the case of complements of hyperplanes.

The defect of toral Laplace eigenfunctions and arithmetic random waves

We study the defect (or ‘signed area’) distribution of standard toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, either for deterministic eigenfunctions averaged w.r.t. the spatial variable, or in a random Gaussian scenario (‘arithmetic random waves’). In either case we exploit the associated symmetry of the eigenfunctions to show that the expectation (spatial or Gaussian) vanishes. In the deterministic setting, we prove that the variance of the defect of flat eigenfunctions, restricted to balls shrinking above the Planck scale, vanishes for ‘most’ energies. Hence the defect of eigenfunctions restricted to most of the said balls is small. We also construct ‘esoteric’ eigenfunctions with large defect variance, by choosing our eigenfunctions so that to mimic the situation on the hexagonal torus, thus breaking the symmetries associated to the standard torus. In the random Gaussian setting, we establish various upper and lower bounds for the defect variance w.r.t. the Gaussian probability measure. A crucial ingredient in the proof of the lower bound is the use of Schmidt’s subspace theorem.

The Ru–Vojta result for subvarieties

In M. Ru and P. Vojta, A birational Nevanlinna constant and its consequences, Amer. J. Math. 142(3) (2020) 957–991, among other things, proved the so-called general theorem (arithmetic part) which can be viewed as an extension of Schmidt’s subspace theorem. In this paper, we extend their result by replacing the divisors by closed subschemes.

A generalized subspace theorem for closed subschemes in subgeneral position

In this paper, we extend the recent theorem of G. Heier and A. Levin [HL21] 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 (see [Quang19]).

Greatest common divisors of integral points of numerically equivalent divisors

We generalize the G.C.D. results of Corvaja--Zannier and Levin on $\mathbb G_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen-Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman's G.C.D. estimate as an application of Vojta's conjecture, which is substituted by a more general version of Schmidt's subspace theorem of Ru--Vojta in our proof.

A note on some applications of the subspace theorem

A note on some applications of the subspace theorem

Spectral Decompositions Arising from Atzmon’s Hyperinvariant Subspace Theorem

By means of a weaker functional model, we prove the existence of non-trivial closed hyperinvariant subspaces for linear bounded operators generalizing, in particular, a classical theorem of Atzmon and revealing the spectral nature of the hyperinvariant subspaces involved. As an application, we show non-trivial spectral subspaces for Bishop operators on $$L^p[0,1)$$ , $$1\le p<\infty $$ , as long as they satisfy Atzmon’s Theorem, providing, in turns, a local spectral decomposition.

An Amir–Cambern theorem for subspaces of Banach lattice-valued continuous functions

For \(i=1,2\), let \(E_i\) be a reflexive Banach lattice over \({\mathbb {R}}\) with a certain parameter \(\lambda ^+(E_i)>1\), let \(K_i\) be a locally compact (Hausdorff) topological space and let \({\mathcal {H}}_i\) be a closed subspace of \({\mathcal {C}}_0(K_i, E_i)\) such that each point of the Choquet boundary \({\text {Ch}}_{{\mathcal {H}}_i} K_i\) of \({\mathcal {H}}_i\) is a weak peak point. We show that if there exists an isomorphism \(T:{\mathcal {H}}_1\rightarrow {\mathcal {H}}_2\) with \(\left\| T\right\| \cdot \left\| T^{-1}\right\| <\min \lbrace \lambda ^+(E_1), \lambda ^+(E_2) \rbrace\) such that T and \(T^{-1}\) preserve positivity, then \({\text {Ch}}_{{\mathcal {H}}_1} K_1\) is homeomorphic to \({\text {Ch}}_{{\mathcal {H}}_2} K_2\).

A generalized Schmidt subspace theorem for closed subschemes

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 extending the scope of the use of Seshadri constants in this context in a natural way.

On Invariant Subspaces of Subdecomposable Operators

In this paper, we prove the Mohebi-Radjabalipour Conjecture under an additional condition, and obtain a new invariant subspace theorem for subdecomposable operators. Our main results contain known results in the literature as special cases. Moreover, we give some specific examples to illustrate that the conditions in our theorem are significantly weaker, simpler and more nature than those in the known results.

On arithmetic inequalities for points of bounded degree

We study algebraic points of bounded degree on polarized projective varieties. To do so, we refine further the filtration construction and Subspace Theorem approach, for the study of integral points, which has origins in the work of Corvaja-Zannier, Levin, Evertse and Autissier. Our main result shows how a conjecture of H.~P.~Schlickewei implies Second Main Arithmetic Schmidt's Subspace type inequalities for polarized projective varieties and points of bounded degree.

A Khintchine-type theorem for affine subspaces

We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parameterizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine’s theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of [Formula: see text] intersected with an appropriate subspace is also obtained.

Another 𝑆-unit variant of Diophantine tuples

We show that there are only finitely many triples of integers 0 &gt; a &gt; b &gt; c 0 &gt; a &gt; b &gt; c such that the product of any two of them is the value of a given polynomial with integer coefficients evaluated at an S S -unit that is also a positive integer. The proof is based on a result of Corvaja and Zannier and thus is ultimately a consequence of the Schmidt subspace theorem.

On iterated product sets with shifts, II

The main result of this paper is the following: for all b ∈ ℤ there exists k = k ( b ) such that max { | A ( k ) | , | ( A + u ) ( k ) | } ≥ | A | b , for any finite A ⊂ ℚ and any nonzero u ∈ ℚ . Here, | A ( k ) | denotes the k -fold product set { a 1 ⋯ a k : a 1 , … , a k ∈ A } . Furthermore, our method of proof also gives the following l ∞ sum-product estimate. For all γ > 0 there exists a constant C = C ( γ ) such that for any A ⊂ ℚ with | A A | ≤ K | A | and any c 1 , c 2 ∈ ℚ ∖ { 0 } , there are at most K C | A | γ solutions to c 1 x + c 2 y = 1 , ( x , y ) ∈ A × A . In particular, this result gives a strong bound when K = | A | 𝜖 , provided that 𝜖 > 0 is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem. In further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products. We utilize a query-complexity analogue of the polynomial Freiman–Ruzsa conjecture, due to Palvolgyi and Zhelezov (2020). This new tool replaces the role of the complicated setup of Bourgain and Chang (2004), which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.

Greatest common divisors with moving targets and consequences for linear recurrence sequences

We establish consequences of the moving form of Schmidt’s Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving S S -unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin, and others.

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and E(Fq) be an elliptic curve over Fq given by the equation y2=x3+Ax+B. For a positive integer n we denote by #E(Fqn) the number of rational points on E (including infinity) over the extension Fqn. Under a mild technical condition, we show that the sequence {#E(Fqn)}n>0 contains at most 10200 perfect squares. If the mild condition is not satisfied, then #E(Fqn) is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range q<50 and n≤1000.

On the degeneracy of integral points and entire curves in the complement of nef effective divisors

As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a conjecture regarding bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.