An analogue of the convergence part of the Khintchine-Groshev theorem, as well as its multiplicative version, is proved for nondegenerate smooth submanifolds in $\mathbb{R}^n$. The proof combines methods from metric number theory with a new approach involving the geometry of lattices in Euclidean spaces.
We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parametrizing 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 W(τ) intersected with an appropriate subspace is also obtained.
In the paper we provide measure estimates for the set of numbers whose sequence of products of continued fraction partial quotients $M_n = a_1 \ldots a_n$ has exponential growth with rate close to the one predicted by Khintchine's theorem, i.e. for which \begin{equation*} e^{(κ- T)n} \leqslant M_n \leqslant e^{(κ+ T)n} \end{equation*} for a fixed $T > 0$ and all $n$ greater than some fixed integer $N$, where $e^κ= 2.685\ldots$ is the Khintchine constant. Choosing $N$ large enough the measure can be made arbitrarily close to full, for any given $T$. The bounds are not of asymptotic nature, but …
We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric theory associated with mixed problems in this setting. This includes finding a natural mixed analogue of the classical Jarník-Besicovich Theorem. Previous knowledge surrounding mixed problems was almost entirely restricted to the multiplicative setup of de Mathan & Teulié [21], where the concept originated.
In this paper we adapt parametric geometry of numbers developed by Wolfgang Schmidt and Leonard Summerer to a multiplicative setting, and derive a chain of inequalities for the corresponding exponents which splits the transference inequality for Diophantine exponents of lattices in the same way Khintchine's transference inequalities for simultaneous approximation can be split.
Khintchine's and Dyson's transference theorems can be very easily deduced from Mahler's transference theorem. In the multiplicative setting an obstacle appears, which does not allow deducing the multiplicative transference theorem immediately from Mahler's theorem. Some extra considerations are required, for instance, induction by the dimension. In this paper we propose an analogue of Mahler's theorem which implies the multiplicative transference theorem immediately.
In this paper we establish the convergence case of Khintchine's theorem for affine hyperplanes in function field of positive characteristic. Along with that, we also prove a quantitative version of the same. The main technique used in the proof is a dynamical result called `Quantitative nondivergence' due to D. Y. Kleinbock and G. A. Margulis [17].
In his 1960 paper, Schmidt studied a quantitative type of Khintchine-Groshev theorem for general (higher) dimensions. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more generally, to add on the congruence condition by M. Alam, A. Ghosh, and S. Yu. In this paper, we generalize this new approach to S-arithmetic spaces and obtain a quantitative version of an S-arithmetic Khintchine-Groshev theorem. In fact, we consider a new S-arithmetic analog of Diophantine approximation, which is different from the one formerly established (see t…
In 1962, Gallagher proved an higher dimensional version of Khintchine's theorem on Diophantine approximation. Gallagher's theorem states that for any non-increasing approximation function $ψ:\mathbb{N}\to (0,1/2)$ with $\sum_{q=1}^{\infty} ψ(q)\log q=\infty$ and $γ=γ'=0$ the following set \[ \{(x,y)\in [0,1]^2: \|qx-γ\|\|qy-γ'\|<ψ(q) \text{ infinitely often}\} \] has full Lebesgue measure. Recently, Chow and Technau proved a fully inhomogeneous version (without restrictions on $γ,γ'$) of the above result. In this paper, we prove an Erdős-Vaaler type result for fibred multiplicative Diophantine…
We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin-Schaeffer Conjecture (1941), and are likely to be more accessible. We point out that the direct random analogue of the Duffin-Schae…
The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in $\mathbb R^m$ to approximation of systems of linear forms in $\mathbb R^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine-Groshev theorem which does not carry a monotonicity assumption when $nm>2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani (2010) that monotonicity is not required when $nm>1$. That result resolve…
The Duffin--Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine's Theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine's Theorem studied. We prove versions of Khintchine's T…
There are two main interrelated goals of this paper. Firstly we investigate the sums \[ S_N(α,γ):=\sum_{n=1}^N\frac{1}{n\|nα-γ\|}~~~\text{and}~~~ R_N(α,γ):=\sum_{n=1}^N\frac{1}{\|nα-γ\|}\,, \] where $α$ and $γ$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W. M. Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of \[ \{1\le n\le N:\|nα-γ\|<\varepsilon\} \, , \] valid for all sufficiently large $N$ and all sufficiently small …
In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.
In this paper we consider a multiparametric version of Wolfgang Schmidt and Leonard Summerer's parametric geometry of numbers. We apply this approach in two settings: the first one concerns weighted Diophantine approximation, the second one concerns Diophantine exponents of lattices. In both settings we use multiparametric approach to define intermediate exponents. Then we split the weighted version of Dyson's transference theorem and an analogue of Khintchine's transference theorem for Diophantine exponents of lattices into chains of inequalities between the intermediate exponents we define b…
We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure $\mathbb{R}_{\exp}$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The first one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts, and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of t…
Cobham's theorem asserts that if a sequence is automatic with respect to two multiplicatively independent bases, then it is ultimately periodic. We prove a stronger density version of the result: if two sequences which are automatic with respect to two multiplicatively independent bases coincide on a set of density one, then they also coincide on a set of density one with a periodic sequence. We apply the result to a problem of Deshouillers and Ruzsa concerning the least nonzero digit of $n!$ in base $12$.
In this paper we prove an upper bound on the "size" of the set of multiplicatively $ψ$-approximable points in $\mathbb R^d$ for $d>1$ in terms of $f$-dimensional Hausdorff measure. This upper bound exactly complements the known lower bound, providing a "zero-full" law which relates the Hausdorff measure to the convergence/divergence of a certain series in both the homogeneous and inhomogeneous settings. This zero-full law resolves a question posed by Beresnevich and Velani (2015) regarding the "log factor" discrepancy in the convergent/divergent sum conditions of their theorem. We further prov…
We generalize two results about subgroups of multiplicative group of finite field of prime order. In particular, the lower bound on the cardinality of the set of values of polynomial $P(x,y)$ is obtained under the certain conditions, if variables $x$ and $y$ belong to a subgroup $G$ of the multiplicative group of the filed of residues. Also the paper contains a proof of the result that states that if a subgroup $G$ can be presented as a set of values of the polynomial $P(x,y)$, where $x\in A$, and $y\in B$ then the cardinalities of sets $A$ and $B$ are close (in order) to a square root of the …
We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear as common factors of two automatic sequences defined over multiplicatively independent bases. This generalises the famous theorem of Cobham.
In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.
We prove a sharp version of Halász's theorem on sums $\sum_{n \leq x} f(n)$ of multiplicative functions $f$ with $|f(n)|\le 1$. Our proof avoids the "average of averages" and "integration over $α$" manoeuvres that are present in many of the existing arguments. Instead, motivated by the circle method we express $\sum_{n \leq x} f(n)$ as a triple Dirichlet convolution, and apply Perron's formula.
We construct a version of Beilinson's regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic K-theory. We use this theory to give an interpretation of Bloch's construction of K_3-classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic K-theory of number rings.
