Mahbub Alam, Anish Ghosh, Shucheng Yu
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.
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We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.
We prove two results concerning the generalized Fermat equation $x^4+y^4=z^p$. In particular we prove that the First Case is true if $p \neq 7$.
For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13, 17, 19, 23 \pmod{24}$. In this paper, we investigate the parity of $b_{21}(n)$ involving the primes $p$ with $p\equiv 1, 5, 7, 11 \pmod{24}$. We prove new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving the odd pr…
Infinite families of Diophantine quadruples in $\mathbb{Z}[\sqrt{-2}]$ in the remaining exceptional congruence classes
研究$\mathbb{Z}[\sqrt{-2}]$中剩余例外同余类的丢番图四元组，通过结合正则扩展方法与新族构造明确的$D(z)$-四元组，证明每个例外类含无穷多$z$值可形成两次半正则的$D(z)$-四元组。
We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.
We give bounds on the number of solutions to the Diophantine equation (X+1/x)(Y+1/y) = n as n tends to infinity. These bounds are related to the number of solutions to congruences of the form ax+by = 1 modulo xy.
Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$. In this paper, we introduce new infinite Ramanujan type congruences modulo $2$ for $b_3(2n)$. They complement naturally the results of Keith-Zanello and Yao and involve primes in $\mathcal P=\{p \text{ prime …
Class fields and form class groups for solving certain quadratic Diophantine equations
本文利用 Shimura 典范模型构造与形式类群对应的类域，并据此给出形如 x²+ny² 且带附加条件的素数刻画，同时推导出高阶层模函数特殊值的 Kronecker 型同余关系。
The paper shows that the asymptotic density of solutions of Diophantine equations or systems of the natural numbers is 0. The author provides estimation methods and estimates number, density and probability of k- tuples $<x_1,...x_k>$ to be the solution of the algebraic equations of the first, second and higher orders in two or more variables, non-algebraic Diophantine equations and systems of Diophantine equations in the domain of the natural numbers. The estimate for the number of positive integer solutions of the second-order Diophantine equations in two, three or more variables is geometri…
Some new results in quantitative Diophantine approximation
本文针对对角三元不定形式的一参数族，建立了两个定量丢番图逼近的新结果，分别涉及素数点上的二次形式与整数点上的任意次数非齐次形式。
Two Central limit theorems in Diophantine approximation
本文沿用 Björklund 与 Gorodnik 的思路，利用累积量方法和齐性空间动力学，分别为带同余条件及非齐次丢番图逼近建立了中心极限定理。
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…
We consider a system of integer polynomials of the same degree with non-singular local zeros and in many variables. Generalising the work of Birch (1962) we find quantitative asymptotics (in terms of the maximum of the absolute value of the coefficients of these polynomials) for the number of integer zeros of this system within a growing box. Using a quantitative version of the Nullstellensatz, we obtain a quantitative strong approximation result, i.e. an upper bound on the smallest integer zero provided the system of polynomials is non-singular.
We consider multiparameter dynamics on the space of unimolular lattices. Along with quantitative nondivergence we prove that multiplicative Diophantine exponents of hyperplanes are inherited by their nondegenerate submanifolds.
We present a sharpening of nondivergence estimates for unipotent (or more generally polynomial-like) flows on homogeneous spaces. Applied to metric Diophantine approximation, it yields precise formulas for Diophantine exponents of affine subspaces of $\Bbb R^n$ and their nondegenerate submanifolds.
We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.
Effective Results in The Metric Theory of Quantitative Diophantine Approximation
本文将丢番图逼近度量理论中依赖隐式常数的渐近定量化为有效结果，在测度至多 δ 的例外集外显式控制常数，并推出 Schmidt 定理、Koukoulopoulos–Maynard 定理及正规数等结果。
Recently, Adiceam, Beresnevich, Levesley, Velani and Zorin proved a quantitative version of the convergence case of the Khintchine-Groshev theorem for nondegenerate manifolds, motivated by applications to interference alignment. In the present paper, we obtain analogues of their results for affine subspaces.
The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational points lying close to manifolds and badly approximable points on manifolds. The main emphasis is on the role of the Quantitative non-Divergence estimate in the aforementioned topics within the theory of Diophantine approximation, and therefore this paper should not be regarded as a comprehensive overview of the area.
Geometry of Diophantine exponents
本综述探讨了丢番图指数及其满足的关系，介绍了经典丢番图指数、带权与乘性类似物及格的指数，并重点关注转换原理。
Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers
本文在具有傅里叶衰减条件的集合上建立了分母受缺项序列限制的定量非齐次 Khintchine 型定理，改进并推广了已有结果，并证明 Dirichlet 不可改进集与良逼近集具有正傅里叶维数。
On the quantitative variation of congruence ideals and integral periods of modular forms
本文证明了 Pollack–Weston 关于高权模形式 Ribet 式降级同余的定量猜想，方法依托 $R=\mathbb{T}$ 定理与伴随 $L$ 值比较，并指出其算术应用。
In the present paper we establish bounds for the size of the spectral gap for actions of algebraic groups on certain homogeneous spaces. Our approach is based on estimating operator norms of suitable averaging operators, and we develop techniques for establishing both upper and lower bounds for such norms. We shall show that this analytic problem is closely related to the arithmetic problem of establishing bounds on the discrepancy of distribution for rational points on algebraic group varieties. As an application, we show how to establish an effective bound for property $τ$ of congruence subg…
Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets
本文在矩阵设定下证明了分形集上的定量Schmidt型计数定理，扩展了先前关于丢番图逼近的研究，并增强了序列的等分布性质，应用于向量正规性与收缩目标问题。
Non-degeneracy and Fibering with Applications to Diophantine Approximation
本文针对ℝⁿ中的非退化流形开发新纤维化机制，证明其可分解为具一致定量非退化性质的曲线族，该结构在小Cⁿ微扰下稳定，为丢番图逼近结果从曲线扩展到流形提供框架。
In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a consequence of this result for polynomials structures in sumsets and limitations of the method.
Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations
本文通过沿幂单方向的定量回复方法，给出了高阶对角化作用在 $SL_2$ 形式算术商上正熵测度的分类及其 Adelic 版本，并应用于整数倍丢番图逼近问题。
Transcendence and measures via the refined Diophantine exponent
引入精细丢番图指数，检测被噪声部分掩盖的重复模式，统一并扩展了Adamczewski-Bugeaud及Corvaja-Zannier等人的超越性结果，并发展了超越测度的定量细化。
An open conjecture of Erdos and Moser is that the only solution of the Diophantine equation in the title is the trivial solution 1+2=3. Reducing the equation modulo k and k^2, we give necessary and sufficient conditions on solutions to the resulting congruence and supercongruence. A corollary is a new proof of Moser's result that the conjecture is true for odd exponents n. We also connect solutions k of the congruence to primary pseudoperfect numbers and to a result of Zagier. The proofs use divisibility properties of power sums as well as Lerch's relation between Fermat and Wilson quotients.
Let $F \subseteq [0,1]$ be a set that supports a probability measure $μ$ with the property that $ |\widehatμ(t)| \ll (\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A >2$, we establish a quantitative inhomogeneous Khintchine-type theorem in which (i) the points of interest are restricted to $F$ and (ii) the denominators of the `shifted' rationals are restricted to $\mathcal{A}$. The theorem can be viewed as a natural strengthening of the fact that the sequence $(q_nx {\rm \ mod \, } 1)…
We generalize Dirichlet's diophantine approximation theorem to approximating any real number $α$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be related to the congruence equation problem $x y \equiv c \pmod q$ with $1 \leq x, y \leq q^{1/2 + ε}$.
The distribution of $αp$ modulo one, where $p$ runs over the rational primes and $α$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $ν>0$ one can establish the infinitude of primes $p$ satisfying $||αp||\le p^{-ν}$. The latest record in this regard is Kaisa Matomäki's landmark result $ν=1/3-\varepsilon$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau and the first-named author, investigated the same problem in the context of imaginary quadratic fields. Glyn Harman obtained an analog…
We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $α$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in …
We consider the problem of Diophantine approximation on semisimple algebraic groups by rational points with restricted numerators and denominators and establish a quantitative approximation result for all real points in the group by rational points with any prescribed denominator and an almost prime numerator.
Diophantine approximation with sums of two squares II
本文研究了二平方和在丢番图逼近中的定量版本，针对无理数 α 和正定二次型 Q，证明了存在无限多个 n 满足特定不等式条件。
On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications
本文证明两个无复乘新形式在素数同余下导致对应阿贝尔簇自同态代数同构，并应用于证明特定丢番图方程无解。
Automorphic density estimates and optimal Diophantine exponents
本文利用四元数代数导出的一致不可约算术同余格点新近建立的精细Sarnak密度假设，无条件建立了其在对称空间上作用的最优逼近指数，并给出了任意Gelfand对乘积中格点逼近指数最优性的一般谱准则。
This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.
A minimalist version of the circle method and Diophantine problems over thin sets
本文提出圆法的极简版本，证明均值估计、Weyl 型估计及容许剩余类上的分布性质即可建立华林问题等加性问题的渐近公式，并将其推广到椭圆薄集等薄集上。
In a ground-breaking work \cite{BY}, Beresnevich and Yang recently proved Khintchine's theorem in simultaneous Diophantine approximation for nondegenerate manifolds resolving a long-standing problem in the theory of Diophantine approximation. In this paper, we prove an effective version of their result.
An asymptotic formula which holds almost everywhere is obtained for the number of solutions to the Diophantine inequalities |qA-p|<ψ(|q|), where A is an n by m matrix (m>1) over the field of formal Laurent series with coefficients from a finite field, and p and q are vectors of polynomials over the same finite field.
Diophantine approximation with sums of two squares
研究二元二次型在丢番图逼近中的应用，证明对于任意无理数α，存在无限多个由Q表示的正整数n，满足||αn||<n^(-1/2-ε)。同时给出当指数1/2-ε替换为任意固定γ<3/7时的结果量化版本。
Quantitative level lowering for weight two Hilbert modular forms
该研究将Ribet和Takahashi的椭圆曲线Shimura曲线参数化结果推广到希尔伯特模情形，得到特定Shimura曲线上上同调同余模等于环论同余模的结论，扩展了相关学者的研究成果。
Quantitative time-averaged spherical means along equidistributed spirals: Diophantine rates and limits of uniformity
该研究针对Kronecker流生成的等分布螺旋，证明了球面可观测量时间平均的多项式收敛速率，处理了参数化退化问题，还得到收缩螺旋下的显式速率及径向剖面二分性，并指出无统一速率适用于所有有理独立频率。
Quantitative inhomogeneous Diophantine approximation for systems of linear forms
本文在非齐次参数为非Liouville无理数时，证明了$(m,n)=(1,2)$情形下非齐次Khintchine-Groshev定理无需单调性假设，并进一步得到了具有几乎平方根抵消的渐近公式。
The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and $p$-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of $\Bbb R^n$) are generalized to the $S$-arithmetic setting.
This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for non-degenerate submanifolds of $\mathbb{R}^n$. The importance of such quanti…
The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong extremality' of arbitrary finite collection of smooth nondegenerate submanifolds of ${\bold R}^n$. The proofs are based on generalized quantitative nondivergence estimates for translates of measures on the space of lattices.
Let $Λ$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / Λ$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong approximation result for Zariski-dense $Λ$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).
We present shrinking targets results for general systems with the emphasis on applications for IETs (interval exchange transformations) $(J,T)$, $J=[0,1)$. In particular, we prove that if an IET $(J,T)$ is ergodic (relative to the Lebesgue measure $\lam$), then the equality \[ \liminf_{n\to\infty}\limits n |T^n(x)-y|=0 \tag{A1} \] holds for $\lam\ttimes\lam$-a. a. $(x,y)\in J^2$. The ergodicity assumption is essential: the result does not extend to all minimal IETs. The factor $n$ in (A1) is optimal (e. g., it cannot be replaced by $n \ln(\ln(\ln n))$. On the other hand, for Lebesgue almost al…
Let $Y_0$ be a not very well approximable $m\times n$ matrix, and let $M$ be a connected analytic submanifold in the space of $m\times n$ matrices containing $Y_0$. Then almost all $Y\in M$ are not very well approximable. This and other similar statements are cast in terms of properties of certain orbits on homogeneous spaces and deduced from quantitative nondivergence estimates for `quasi-polynomial' flows on on the space of lattices.
Quantitative Khintchine on the parabola with non-monotonic approximation functions
该研究针对抛物线上的点，证明了度量丢番图逼近中Khintchine定理收敛情形的定量版本，核心是突破了逼近函数需单调的限制，还用到了Burgess界等经典数论结果。
In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function $ψ:\mathbb{N}\to (0,1/2)$ with $\sum_q ψ(q)=\infty$ and any number $γ,$ the following set \[ W(ψ,γ)=\{x\in [0,1]: |qx-p-γ|< ψ(q) \text{ for infinitely many } q,p\in\mathbb{N}\} \] has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if $γ$ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced …
Point counting estimates are a key stepping stone to various results in metric Diophantine approximation. In this paper we use the quantitative non-divergence estimates originally developed by Kleinbock and Margulis to improve lower bounds by Bernik, Götze et al. for the number of points with algebraic conjugate coordinates close to a given manifold. In the process, we also improve on a Khinchin-Groshev-type theorem for a problem of constrained approximation by polynomials.
Moment formulas of Siegel transforms with congruence conditions in dimension 2
本文计算了带同余条件的二维 Siegel 变换的一阶和二阶矩公式，并据此导出了 Schmidt 随机计数定理的类比及对向量施加同余约束的定量 Khintchine 定理。
For a positive proportion of primes $p$ and $q$, we prove that $\mathbb{Z}$ is Diophantine in the ring of integers of $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$. This provides a new and explicit infinite family of number fields $K$ such that Hilbert's tenth problem for $O_K$ is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.
We prove a quantitative theorem for Diophantine approximation by rational points on spheres. Our results are valid for arbitrary unimodular lattices and we further prove 'spiraling' results for the direction of approximates. These results are quantitative generalizations of the Khintchine-type theorem on spheres proved by Kleinbock and Merrill.
Inhomogeneous Diophantine Approximation on $M_0$-sets
本文在 M_0-集上建立了带约束分母的定量 Schmidt 型逼近定理，通过分母增长与 Rajchman 测度傅里叶变换衰减之间的平衡条件，得到非齐次 Khintchine-Järnik 型结果及交集 Hausdorff 维数下界。
On qualitative aspects of the quantitative subspace theorem
本文基于参数化子空间定理，通过引入线性截面和对数扭曲高度函数等工具，推导了数域有限扩张下大线性系统的丢番图算术不等式，并建立了丢番图例外集的新颖线性散射型结果。
Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation
本文为$\mathbb{R}^d$-作用证明了涉及膨胀子流形平均的新遍历定理，给出有效混合假设下的误差项定量估计，并据此研究了实系数矩阵的乘性Dirichlet可改进性，证明了几乎所有矩阵的一致逼近性，提出了Littlewood猜想的强化问题。
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