arXiv subjects

Khoa D. Nguyen

Publications and source records attributed to Khoa D. Nguyen.

33 records · Page 2Linked to original sources

arXiv2016

The Dynamical Manin-Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps

We prove the Dynamical Bogomolov Conjecture for endomorphisms of P^1\times P^1 defined over a number field. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we prove the Dynamical Manin-Mumford Conjecture for endomorhisms of P^1\times P^1 defined over the complex numbers.

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arXiv2016

Squarefree Doubly Primitive Divisors in Dynamical Sequences

Let K be a number field or a function field of characteristic 0, let f be a K-rational function of degree greater than 1, and let a be an element of K. Let S be a finite set of places of K containing all the archimedean ones and the primes where f has bad reduction. After excluding all the natural counter-examples, we define a subset A(f,a) of pairs of integers (m,n) with m nonnegative and n positive, and show that for all but finitely many (m,n) in A(f,a) there is a prime p of K which is not in S such that the p-adic valuation of f^{m+n}(a)-f^m(a) is precisely equal to 1, and moreover a has portrait (m,n) under the action of f modulo p. This latter condition implies that the p-adic valuation of f^{u+v}(a)-f^u(a) is not positive if u is a nonnegative integer and v is a positive integer with u<m or v<n. Our proof assumes a conjecture of Vojta in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram-Silverman, Faber-Granville, and of the authors.

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arXiv2015

Algebraic approximations to linear combinations of powers: an extension of results by Mahler and Corvaja-Zannier

For every complex number $x$, let $\Vert x\Vert_{\mathbb{Z}}:=\min\{|x-m|:\ m\in\mathbb{Z}\}$. Let $K$ be a number field, let $k\in\mathbb{N}$, and let $α_1,\ldots,α_k$ be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of $θ\in (0,1)$ such that there are infinitely many tuples $(n,q_1,\ldots,q_k)$ satisfying $\Vert q_1α_1^n+\ldots+q_kα_k^n\Vert_{\mathbb{Z}}<θ^n$ where $n\in\mathbb{N}$ and $q_1,\ldots,q_k\in K^*$ having small logarithmic height compared to $n$. In the special case when $q_1,\ldots,q_k$ have the form $q_i=qc_i$ for fixed $c_1,\ldots,c_k$, our work yields results on algebraic approximations of $c_1α_1^n+\ldots+c_kα_k^n$ of the form $\displaystyle \frac{m}{q}$ with $m\in \mathbb{Z}$ and $q\in K^*$ (where $q$ has small logarithmic height compared to $n$). Various results on linear recurrence sequences also follow as an immediate consequence. The case $k=1$ and $q_1$ is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. The use of the Subspace Theorem based on work of Corvaja-Zannier together with several modifications play an important role in the proof of our results.

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arXiv2015

Density Evolution Analysis of Spatially Coupled LDPC Codes Over BIAWGN Channel

In this paper, we study the density evolution analysis of spatially coupled low-density parity-check (SC-LDPC) codes over binary input additive white Gaussian noise (BIAWGN) channels under the belief propagation (BP) decoding algorithm. Using reciprocal channel approximation and Gaussian approximation, we propose averaging techniques for the density evolution of SC-LDPC codes over BIAWGN channels. We show that the proposed techniques can closely predict the decoding threshold while offering reduced complexity compared to the existing multi-edge-type density evolution.

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arXiv2015

Dynamics of split polynomial maps: uniform bounds for periods and applications

Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree d > 1 is said to be disintegrated if neither f nor -f is linearly conjugate to x^d or T_d(x) where T_d is the Chebyshev polynomial of degree d. Let m and n be integers greater than 1, we prove that there exists an effectively computable constant c(m,n) depending only on m and n such that the following holds. Let f_1,...,f_n be polynomials with coefficients in K, which are disintegrated polynomials of degree at most m and let F be the induced coordinate-wise self-map of the n-th dimensional affine space, i.e. F(x_1,..,x_n)=(f_1(x_1),...,f_n(x_n)). Then the period of every irreducible F-periodic subvariety of the n-th dimensional affine space with non-constant projection to each coordinate axis is at most c(m,n). As an immediate application, we prove an instance of the dynamical Mordell-Lang problem following recent work of Xie. The main technical ingredients are Medvedev-Scanlon classification of invariant subvarieties together with classical and more recent results in Ritt's theory of polynomial decomposition.

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arXiv2015

Some finiteness results on monogenic orders in positive characteristic

This work is motivated by the papers [EG85] and [Ngu15] in which the following two problems are solved. Let $\mathcal{O}$ is a finitely generated $\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero, consider the following problems: (A) Fix $s$ that is integral over $\mathcal{O}$, describe all $t$ such that $\mathcal{O}[s]=\mathcal{O}[t]$. (B) Fix $s$ and $t$ that are integral over $\mathcal{O}$, describe all pairs $(m,n)\in\mathbb{N}^2$ such that $\mathcal{O}[s^m]=\mathcal{O}[t^n]$. In this paper, we solve these problems and provide a uniform bound for a certain "discriminant form equation" that is closely related to Problem (A) when $\mathcal{O}$ has characteristic $p>0$. While our general strategy roughly follows [EG85] and [Ngu15], many new delicate issues arise due to the presence of the Frobenius automorphisms $x\mapsto x^p$. Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this paper.

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arXiv2015

On modules of integral elements over finitely generated domains

This paper is motivated by the results and questions of Jason P. Bell and Kevin G. Hare in the paper "On $\mathbb{Z}$-modules of algebraic integers" (Canad. J. Math. Vol. 61, 2009). Let $\mathcal{O}$ be a finitely generated $\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero. We investigate the following two problems: (A) Fix $q$ and $r$ that are integral over $\mathcal{O}$, describe all pairs $(m,n)\in\mathbb{N}^2$ such that $\mathcal{O}[q^m]=\mathcal{O}[r^n]$. (B) Fix $r$ that is integral over $\mathcal{O}$, describe all $q$ such that $\mathcal{O}[q]=\mathcal{O}[r]$. In this paper, we solve Problem (A), present a solution of Problem (B) by Evertse and Győry, and explain their relation to the paper of Bell and Hare. In the following, $c_1$ and $c_2$ are effectively computable constants with a very mild dependence on $\mathcal{O}$, $q$, and $r$. For (B), Evertse and Győry show that there are $N\leq c_2$ elements $s_1,\ldots,s_N$ such that $\mathcal{O}[s_i]=\mathcal{O}[r]$ for every $i$, and for every $q$ such that $\mathcal{O}[q]=\mathcal{O}[r]$, we have $q-us_i\in\mathcal{O}$ for some $1\leq i\leq N$ and $u\in\mathcal{O}^*$. This immediately answers two questions about Pisot numbers by Bell and Hare in ibid. For (A), we show that except some "degenerate" cases that can be explicitly described, there are at most $c_1$ such pairs $(m,n)$. This significantly strengthens some results in ibid. We also make some remarks on effectiveness and discuss further questions at the end of the paper.

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arXiv2014

Causal/Predictive Imperfect Channel State Information in Block-Fading Channels

We consider a multi-input multi-output (MIMO) block-fading channel with a general model for channel state information at the transmitter (CSIT). The model covers systems with causal CSIT, where only CSIT of past fading blocks is available, and predictive CSIT, where CSIT of some future fading blocks is available. The optimal diversity-multiplexing tradeoff (DMT) and rate-diversity tradeoff (RDT) of the channel are studied under long-term power constraints. The impact of imperfect (mismatched) CSIT on the optimal DMT and RDT is also investigated. Our results show the outage diversity gain obtained by providing imperfect causal/predictive CSIT, leading to new insights into system design and analysis.

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arXiv2014

Delay-Exponent of Bilayer Anytime Code

In this paper, we study the design and the delay-exponent of anytime codes over a three terminal relay network. We propose a bilayer anytime code based on anytime spatially coupled low-density parity-check (LDPC) codes and investigate the anytime characteristics through density evolution analysis. By using mathematical induction technique, we find analytical expressions of the delay-exponent for the proposed code. Through comparison, we show that the analytical delay-exponent has a close match with the delay-exponent obtained from numerical results.

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arXiv2013

Finite Length Analysis of LDPC Codes

In this paper, we study the performance of finite-length LDPC codes in the waterfall region. We propose an algorithm to predict the error performance of finite-length LDPC codes over various binary memoryless channels. Through numerical results, we find that our technique gives better performance prediction compared to existing techniques.

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arXiv2010

MIMO ARQ with Multi-bit Feedback: Outage Analysis

We study the asymptotic outage performance of incremental redundancy automatic repeat request (INR-ARQ) transmission over the multiple-input multiple-output (MIMO) block-fading channels with discrete input constellations. We first show that transmission with random codes using a discrete signal constellation across all transmit antennas achieves the optimal outage diversity given by the Singleton bound. We then analyze the optimal SNR-exponent and outage diversity of INR-ARQ transmission over the MIMO block-fading channel. We show that a significant gain in outage diversity is obtained by providing more than one bit feedback at each ARQ round. Thus, the outage performance of INR-ARQ transmission can be remarkably improved with minimal additional overhead. A suboptimal feedback and power adaptation rule, which achieves the optimal outage diversity, is proposed for MIMO INR-ARQ, demonstrating the benefits provided by multi-bit feedback.

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arXiv2008

Power Allocation for Fading Channels with Peak-to-Average Power Constraints

Power allocation with peak-to-average power ratio constraints is investigated for transmission over Nakagami-m fading channels with arbitrary input distributions. In the case of delay-limited block-fading channels, we find the solution to the minimum outage power allocation scheme with peak-to-average power constraints and arbitrary input distributions, and show that the signal-to-noise ratio exponent for any finite peak-to-average power ratio is the same as that of the peak-power limited problem, resulting in an error floor. In the case of the ergodic fully-interleaved channel, we find the power allocation rule that yields the maximal information rate for an arbitrary input distribution and show that capacities with peak-to-average power ratio constraints, even for small ratios, are very close to capacities without peak-power restrictions.

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arXiv2007

A Tight Lower Bound to the Outage Probability of Discrete-Input Block-Fading Channels

In this correspondence, we propose a tight lower bound to the outage probability of discrete-input Nakagami-m block-fading channels. The approach permits an efficient method for numerical evaluation of the bound, providing an additional tool for system design. The optimal rate-diversity trade-off for the Nakagami-m block-fading channel is also derived and a tight upper bound is obtained for the optimal coding gain constant.

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arXiv2007

Power Allocation for Discrete-Input Non-Ergodic Block-Fading Channels

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arXiv2007

Power Allocation for Discrete-Input Delay-Limited Fading Channels

We consider power allocation algorithms for fixed-rate transmission over Nakagami-m non-ergodic block-fading channels with perfect transmitter and receiver channel state information and discrete input signal constellations, under both short- and long-term power constraints. Optimal power allocation schemes are shown to be direct applications of previous results in the literature. We show that the SNR exponent of the optimal short-term scheme is given by m times the Singleton bound. We also illustrate the significant gains available by employing long-term power constraints. In particular, we analyze the optimal long-term solution, showing that zero outage can be achieved provided that the corresponding short-term SNR exponent with the same system parameters is strictly greater than one. Conversely, if the short-term SNR exponent is smaller than one, we show that zero outage cannot be achieved. In this case, we derive the corresponding long-term SNR exponent as a function of the Singleton bound. Due to the nature of the expressions involved, the complexity of optimal schemes may be prohibitive for system implementation. We therefore propose simple sub-optimal power allocation schemes whose outage probability performance is very close to the minimum outage probability obtained by optimal schemes. We also show the applicability of these techniques to practical systems employing orthogonal frequency division multiplexing.

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