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Jing-Jing Huang

Publications and source records attributed to Jing-Jing Huang.

14 recordsLinked to original sources
arXiv2024

Extremal affine subspaces and Khintchine-Jarník type theorems

We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of $\mathbb{R}^n$. We also give an essentially complete classification of all Khintchine type affine subspaces, except for some boundary cases within two logarithmic scales. More general Jarník type theorems are proved as well, sometimes without the monotonicity of the approximation function. These results follow as consequences of our novel estimates for the number of rational points close to an affine subspace in terms of diophantine properties of its defining matrix. Our main tool is the multidimensional large sieve inequality and its dual form.

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arXiv2022

On the connection between the Goldbach conjecture and the Elliott-Halberstam conjecture

In this paper we prove that the binary Goldbach conjecture for sufficiently large even integers would follow under the assumption that both the Elliott-Halberstam conjecture and a variant of the Elliott-Halberstam conjecture twisted by the Möbius function hold, in which the sum of their levels of distribution exceeds 1. This continues the work of Pan in 1982. An analogous result for the twin prime conjecture is obtained by Ram Murty and Vatwani in 2017.

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arXiv2020

The density of rational points near hypersurfaces

We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre's Dimension Growth Conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we obtain an optimal Jarník type theorem for simultaneous approximation on hypersurfaces.

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arXiv2020

Simultaneous approximation on affine subspaces

We solve the convergence case of the generalized Baker-Schmidt problem for simultaneous approximation on affine subspaces, under natural diophantine type conditions. In one of our theorems, we do not require monotonicity on the approximation function. In order to prove these results, we establish asymptotic formulae for the number of rational points close to an affine subspace. One key ingredient is a sharp upper bound on a certain sum of reciprocals of fractional parts associated with the matrix defining the affine subspace.

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arXiv2020

On two lattice points problems about the parabola

We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the square root cancellation in the context of the parabola, whereas its analogues are wide open conjectures for the circle and the hyperbola. We also obtain essentially sharp upper bounds for the latter lattice points problem. Our proofs utilize techniques in Fourier analysis, quadratic Gauss sums and character sums.

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arXiv2019

On a generalization of a theorem of Popov

In this paper, we obtain sharp estimates for the number of lattice points under and near the dilation of a general parabola, the former generalizing an old result of Popov. We apply Vaaler's lemma and the Erdős-Turan inequality to reduce the two underlying counting problems to mean values of a certain quadratic exponential sums, whose treatment is subject to classical analytic techniques.

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arXiv2018

Diophantine approximation on the parabola with non-monotonic approximation functions

We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarnik type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess's bound on short character sums.

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arXiv2015

Hausdorff theory of dual approximation on planar curves

Ten years ago, Beresnevich-Dickinson-Velani initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in $\mathbb{R}^2$, which represents the first complete theory of its kind for a general class of manifolds.

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arXiv2015

Rational points near planar curves and Diophantine approximation

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich-Zorin in the divergence case.

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arXiv2011

On the exceptional set for binary Egyptian fractions

For fixed integer $a\ge3$, we study the binary Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular the number $E_a(N)$ of $n\le N$ for which the equation has no positive integer solutions in $x, y$. The asymptotic formula $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}}$$ as $N$ goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of the underlying group structure.

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arXiv2011

Mean value theorems for binary Egyptian fractions II

In this article, we continue with our investigation of the Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular its number of solutions $R(n;a)$ for fixed $a$. We prove a couple of mean value theorems for the second moment $(R(n;a))^2$ and from which we deduce $\log R(n;a)$ satisfies a certain Gaussian distribution with mean $\log 3\log\log n$ and variance $(log 3)^2\log\log n$, which is an analog of the classical theorem of Erd\H os and Kac. And finally these results in all suggest that the behavior of $R(n;a)$ resembles the divisor function $d(n^2)$ in various aspects.

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arXiv2011

Mean Value Theorems for Binary Egyptian Fractions

In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.

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