Boris Adamczewski, Yann Bugeaud
Let $\k$ be an arbitrary field. For any fixed badly approximable power series $Θ$ in $\k((X^{-1}))$, we give an explicit construction of continuum many badly approximable power series $Φ$ for which the pair $(Θ, Φ)$ satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series.
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We prove a metrical result on a family of conjectures related to the Littlewood conjecture, namely the original Littlewood conjecture, the mixed Littlewood conjecture of de Mathan and Teulié and a hybrid between a conjecture of Cassels and the Littlewood conjecture. It is shown that the set of numbers satisfying a strong version of all of these conjectures is large in the sense of Hausdorff dimension restricted to the set of badly approximable numbers.
Siegel Zeros and the Hardy-Littlewood Conjecture
本文针对Siegel零点界问题，给出一种比Fei和Jia的条件证明略简化的新论证，重新证明相同结论。
We show how a recent result by Aka and Shapira on the evolution of continued fractions in a fixed quadratic field implies the classic result of de Mathan and Teullié on the Mixed Littlewood
In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011 Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs. In this paper, we find "Skewes numbers" for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture. Moreover, we present several algorithms to compute such numbers.
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient condition for the conjecture to hold. An advantage of our method is that it is effective, in the sense that we know how to construct the eventual solution.
By means of Peres-Schlag's method we prove the existence of real numbers $α, β$ such that $$ \liminf_{q\to \infty} (q\log^2 q)||αq|| ||βq|| > 0.
Let $m$ and $n$ be positive integers with $m,n \geq 2$. The second Hardy-Littlewood conjecture states that the number of primes in the interval $(m,m+n]$ is always less than or equal to the number of primes in the interval $[2,n]$. Based on new explicit estimates for the prime counting function $π(x)$, we give some new ranges in which this conjecture holds.
The main goal of this note is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan-Teulie Conjecture, also known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of our main result is that, for almost every real number α, \liminf_{n\rar\infty}n(\log n)^2|n|_p\|nα\|=0.
For any given real number $α$ with bounded partial quotients, we construct explicitly continuum many real numbers $β$ with bounded partial quotients for which the pair $(α, β)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.
We consider a Laurent series defined by infinite products $g_u(t) = \prod_{n=0}^\infty (1 + ut^{-2^n})$, where $u\in \mathbb{F}$ is a parameter and $\mathbb{F}$ is a field. We show that for all $u\in\mathbb{Q}\setminus\{-1,0,1\}$ the series $g_u(t)$ does not satisfy the $t$-adic Littlewood conjecture. On the other hand, if $\mathbb{F}$ is finite then $g_u(t)\in \mathbb{F}((t^{-1}))$ is either a rational function or it satisfies the $t$-adic Littlewood conjecture.
A note on the $p$-adic inhomogeneous Littlewood conjecture
本文针对p进非齐次Littlewood猜想，在Bugeaud等人2011年解决p进Littlewood猜想度量版本的基础上，建立了完全非齐次推广，并在混合情形下得到Harrap等人2013年结果的部分非齐次推广。
The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each hyperbola by a random amount which depends on the denominator of the corresponding rational. Using a randomized covering argument we prove that, not only is this randomized version of the Littlewood Conjecture true for almost all choices of centers, an even stronger statement with an extra factor of a logarithm also holds.
In this article, we investigate the bound of the valency of the Cayley graphs of the generalized quaternion groups which guarantees to be Ramanujan. As is the cases of the cyclic and dihedral groups in our previous studies, we show that the determination of the bound in a special setting is related to the classical Hardy-Littlewood conjecture for primes represented by a quadratic polynomial.
On a question of Gowers related to Littlewood's conjecture
本文针对Gowers提出的与Littlewood猜想相关的单位立方体点集双曲距离问题，通过显式构造给出解答，证明了若不添加改进则该思路无法用于证明Littlewood猜想。
We study the asymptotic behaviour of the prime pair counting function $π_{2k}(n)$ by the means of the discrete Fourier transform on $\mathbb{Z}/ n\mathbb{Z}$. The method we develop can be viewed as a discrete analog of the Hardy-Littlewood circle method. We discuss some advantages this has over the Fourier series on $\mathbb{R} /\mathbb{Z}$, which is used in the circle method. We show how to recover the main term for $π_{2k}(n)$ predicted by the Hardy-Littlewood Conjecture from the discrete Fourier series. The arguments rely on interplay of Fourier transforms on $\mathbb{Z}/ n\mathbb{Z}$ and o…
The second Hardy-Littlewood conjecture, that $π(x)+π(y) \geq π(x+y)$ for integers $x$ and $y$ with $\min\{x,y\}\geq 2$, was formulated in 1923. It continues to attract attention to this day, almost 100 years later. In 1975 Udrescu proved that this conjecture holds for $(x,y)$ sufficiently large, but without an explicit effective bound on the region of validity. We shall revisit Udrescu's result, modifying it to obtain explicit effective bounds, ultimately proving that the second Hardy-Littlewood conjecture is in fact unconditionally true. Furthermore we note that constraints on the prime count…
We prove the following function field analog of the Hardy-Littlewood conjecture (which generalizes the twin prime conjecture) over large finite fields. Let n,r be positive integers and q an odd prime power. For distinct polynomials a_1, ..., a_r over F_q of degree <n let π(q,n;a) be the number of monic polynomials f over F_q of degree n such that f+a_1, ..., f+a_r are simultaneously irreducible. We prove that π(q,n;a) asymptotically equals q^n/n^r as q tends to infinity on odd prime powers and n,r are fixed (the tuple a1,...,a_r need not be fixed).
We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers between the $2d$ different primes. We may conclude that there exist infinitely many $2d$ different primes including the twin primes in case of $d=1$ because we can give the lower bounds of the existence density for the $2d$ different primes in this algorithm. We also discuss the Hardy-Littlewood conjecture and the Sophie Germain primes.
In this paper we introduce the concept of an infinite loop mod $n$ and discuss the properties that these objects have. In particular, we show that a real number $α$ is a counterexample to the $p$-adic Littlewood Conjecture if and only if there exists some $m\in\mathbb{N}$ such that $p^kα$ is an infinite loop mod $p^m$, for all $k\in\mathbb{N}$. This paper is the first of a two part series, which investigate the link between infinite loops and the $p$-adic Littlewood Conjecture.
Schinzel's Hypothesis (H) was used by Colliot-Thélène and Sansuc, and later by Serre, Swinnerton-Dyer and others, to prove that the Brauer-Manin obstruction controls the Hasse principle and weak approximation on pencils of conics and similar varieties. We show that when the ground field is Q and the degenerate geometric fibres of the pencil are all defined over Q, one can use these methods to obtain unconditional results by replacing Hypothesis (H) with the finite complexity case of the generalised Hardy-Littlewood conjecture recently established by Green, Tao and Ziegler.
Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$, we establish an asymptotic formula for $$\sum_{n\leq x}Λ(n+h_1)\cdots Λ(n+h_k)λ(n+h_{1}')\cdots λ(n+h_{\ell}')$$ for any $0\leq k\leq 2$ and $\ell \geq 0$. Specializing to either $\ell=0$ or $k=0$, we deduce the previously known results on the Hardy--Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to H…
We study the prime pair counting functions $π_{2k}(x),$ and their averages over $2k.$ We show that good results can be achieved with relatively little effort by considering averages. We prove an asymptotic relation for longer averages of $π_{2k}(x)$ over $2k \leq x^θ,$ $θ> 7/12,$ and give an almost sharp lower bound for fairly short averages over $k \leq C \log x,$ $C >1/2.$ We generalize the ideas to other related problems.
Disproof of the uniform Littlewood conjecture
本文通过构造性方法证明了统一利特伍德猜想不成立，展示了反例集具有正豪斯多夫维数，并反驳了p进数利特伍德问题的统一版本及更一般S-算术设置中的弱化版本。
There has been recent interest in a hybrid form of the celebrated conjectures of Hardy-Littlewood and of Chowla. We prove that for any $k,\ell\ge1$ and distinct integers $h_2,\ldots,h_k,a_1,\ldots,a_\ell$, we have $$\sum_{n\leq X}μ(n+h_1)\cdots μ(n+h_k)Λ(n+a_1)\cdotsΛ(n+a_{\ell})=o(X)$$ for all except $o(H)$ values of $h_1\leq H$, so long as $H\geq (\log X)^{\ell+ε}$. This improves on the range $H\ge (\log X)^{ψ(X)}$, $ψ(X)\to\infty$, obtained in previous work of the first author. Our results also generalize from the Möbius function $μ$ to arbitrary (non-pretentious) multiplicative functions.
The $p$-adic Littlewood conjecture (PLC) states that $\liminf_{q\to\infty} q\cdot |q|_p \cdot ||qx|| = 0$ for every prime $p$ and every real $x$. Let $w_{CF}(x)$ be an infinite word composed of the continued fraction expansion of $x$ and let $\mathrm{T}$ be the standard left shift map. Assuming that $x$ is a counterexample to PLC we get several restrictions on limit elements of the sequence $\{\mathrm{T}^n w_{CF}(x)\}_{n\in\mathbb{N}}$. As a consequence we show that for any such limit element $w$ we must have $\lim_{n\to\infty} P(w,n) - n = \infty$ where $P(w,n)$ is a word complexity of $w$. W…
Variants of the Littlewood conjecture, their connection to uniformly distributed sequences, and the exact order of the discrepancy of van der Corput--Kronecker-type sequences
本文利用有限域上 $X$-进 Littlewood 猜想的反例构造 van der Corput–Kronecker 型序列，并确定其差异度精确阶，从而支持一致分布理论中的最优差异度上界猜想。
On the $P(t)$-adic Littlewood conjecture in odd characteristics
本文证明了在奇特征域上，对于任意不可约多项式 $P(t)$，$P(t)$-adic Littlewood 猜想不成立。
The convergence of an alternating series of Erdős, assuming the Hardy--Littlewood prime tuples conjecture
本文针对Erdős提出的某交错级数收敛性的公开问题，结合素数随机筛选模型与Gallagher计算变体，在强版本Hardy–Littlewood素数元组猜想下证明了该级数条件收敛。
This paper demonstrates that from the Cramer's, Hardy-Littlewood's and Bateman-Horn's conjectures (suggest that the probability of a large positive integer being $x$ a prime - $\frac {1} {\ln(x)}$) it follows that the events consisting in a positive integer $x$ being not divisible by different primes are dependent with the ratio $0,5e^γ$ ($γ$ - Euler's constant). In establishing Hardy-Littlewood's and Bateman-Horn's conjectures, their authors followed the first suggestion by another one assuming the independence of the above-mentioned events, which on the basis of the first suggestion and Mert…
The uniform Littlewood conjecture fails on a set of positive Hausdorff dimension
该数学研究证明一致Littlewood猜想不成立，给出两类反例集合的Hausdorff维数下界，与经典Littlewood猜想的反例维数结论形成对比。
In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo absolute value sequence $\mathcal{D}$, we obtain the sharp criterion such that for almost every $α$ the inequality \begin{equation*} |n|_{\mathcal{D}}|nα-p|\leq ψ(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in\N\times \Z$ for a certain one-parameter family of $ψ$. Also under minor condition on pseudo absolute value sequences $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$, we obtain a sharp criterion on general sequence $ψ(n)$ such that for almost every $α$ the inequali…
Estimating lower limit in the $p$-adic Littlewood conjecture
本文验证了 $p$-adic Littlewood 猜想中下极限的上界，对所有实数和小素数成立，其中 $p=2$ 时 $\varepsilon=1/25$，$3\le p\le 29$ 时上界不超过 $1/10$。
We classify the measures on SL (k,R)/SL (k,Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.
Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture
本文基于Hardy-Littlewood猜想，研究连续素数对分别属于两个Piatetski-Shapiro序列的分布，提出计数函数猜想并给出启发式论证，还证明了奇异级数加权平均的命题。
Counterexamples to the $p(t)$-adic Littlewood Conjecture Over Small Finite Fields
本文在特征为5的域上构造了p(t)-进Littlewood猜想的显式反例，并猜想该反例适用于所有特征p≡1 mod 4的域，同时将已有反例推广到特征7和11，借助自动序列与Diophantine逼近相结合的高效算法填补了函数域上该猜想研究的空白。
The effective version of Chebotarev's density theorem under the Generalized Riemann Hypothesis and the Artin conjecture (cf. Iwaniec and Kowalski's Analytic Number Theory, 5.13) involves a numerical invariant of a subset $D$ of a finite group $G$ that we call the Littlewood Complexity of $D$. We study this invariant in detail. Using this study, and a new application of the large sieve, we give improved versions of two standard questions related to Chebotarev: the bound on the first prime in a Frobenian set, and the asymptotic of the set of primes with given Frobenius in an infinite Galois exte…
On the cardinality of the exception set in Littlewood's discrete conjecture
本文证明在Littlewood离散猜想中，对几乎所有$a$，最小乘积的阶为$1/((\ln p)^{d-1}\ln H)$，刻画了例外集的基数。
On the Second Hardy-Littlewood Conjecture
本文研究第二个Hardy-Littlewood猜想，通过将素数计数函数的次可加性与素数定理误差项相联系，无条件改进了次可加性成立的范围，并在假设黎曼猜想下证明了更广范围内的不等式成立。
A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture
本文研究p进Littlewood猜想的假定反例，证明若反例存在则存在基p展开为一致回复的反例，并给出形态词条件下猜想成立的充分条件，特别证明p=2时猜想对所有纯形态词成立。
This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an orbifold with another triangulation. This method is used to show that eventually periodic continued fractions have partial quotients which have exponential growth when iteratively multiplied by $n$, for $n$ any fixed, natural number.
In this paper we study the Mixed Littlewood Conjecture with pseudo-absolute values. We show that if p is a prime and D is a pseudo-absolute value sequence satisfying mild conditions then then the infimum over natural numbers n of the quantity n.|n|_p.|n|_D.||nx|| equals 0 for all real x. Our proof relies on a measure rigidity theorem due to Lindenstrauss and lower bounds for linear forms in logarithms due to Baker and Wustholz. We also deduce the answer to the related metric question of how fast the infimum above tends to zero, for almost every x.
Winning property of counterexamples to Uniform Littlewood's Conjecture
本文证明了BFK25提出的一致Littlewood猜想反例集合是超平面绝对获胜的，且在二维实数空间中具有满Hausdorff维数。
We present a class of lattices in R^d (d >= 2) which we call GL-lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood's conjecture amounts to saying that Z^2 is GL. We then prove existence of GL lattices by first establishing a dimension bound for the set of possible exceptions. Existence of vectors (GL-vectors) in R^d with special Diophantine properties is proved by similar methods. For dimension d >= 3 we give explicit constructions of GL lattices (and in fact a much stronger property). We also show that GLC is implied by a conjecture of G. A. Ma…
A long-standing conjecture of Littlewood about simultaneous Diophantine approximation has an analogous problem for a field of formal Laurent series $\mathbb{F}(\!(t^{-1})\!)$. That is, we can ask whether for any series $Θ$, $Φ$ and any $ε>0$, there is a polynomial $α$ such that $|α|\langleαΘ\rangle\langleαΦ\rangle$ where $\langleΘ\rangle=\underset{β\in\mathbb{F}[t]}{\inf}|Θ-β|$. If the base field $\mathbb{F}$ is infinite, then the answer is negative due to Davenport and Lewis (1963). We give a connection between the combinatorics of an orbit under a semigroup action and Diophantine approximati…
The paper investigates various $p$-adic versions of Littlewood's conjecture, generalizing a set-up considered recently by de Mathan and Teulie. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff dimension zero. The proof follows the measure ridigity approach of Einsiedler, Katok and Lindenstrauss.
We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.
Dispersion and Littlewood's conjecture
本文利用色散估计和三距离定理，构造显式全测度集，显著改进了Littlewood猜想相关不等式的定量界，并证明例外集Fourier维数为零。
Reciprocity of Skew Hall-Littlewood-Schubert Series
本文提出并证明了广义Igusa函数和Hall-Littlewood-Schubert级数的联合泛化与细化，解决了Maglione和Voll提出的问题。
In this paper, we determine the bound of the valency of the odd circulant graph which guarantees to be a Ramanujan graph for each fixed number of vertices. In almost of the cases, the bound coincides with the trivial bound, which comes from the trivial estimate of the largest non-trivial eigenvalue of the circulant graph. As exceptional cases, the bound in fact exceeds the trivial one by two. We then prove that such exceptionals occur only in the cases where the number of vertices has at most two prime factors and is represented by a quadratic polynomial in a finite family and, moreover, under…
We show that, for any $0<γ<1/2$, any $(α,β)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrtγ$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such that $n\langle nα\rangle\langle nβ\rangle<\varepsilon$ is greater than $γ\varepsilon\log N$ up to a uniform constant. This can be seen as a quantitative result on the fact that the exceptional set to Littlewood's conjecture has Hausdorff dimension zero, obtained by M. Einsiedler, A. Katok and E. Lindenstrauss in 2000's. For the proof, we study the behavior of the empirical measure…
For any prime $p$ and real number and $α$, the $p$-adic Littlewood Conjecture due to de Mathan and Teulié asserts that \[\inf_{|m|\ge1}|m|_p\cdot |m|\cdot |\left\langleαm\right\rangle|=0.\] Above, $|m|$ is the usual absolute value, $|m|_p$ is the $p$-adic norm and $\left|\left\langle x\right\rangle\right|$ is the distance from $x\in\mathbb{R}$ to the nearest integer. Let $\mathbb{K}$ be a field and $P(t)\in\mathbb{K}[t]$ be an irreducible polynomial. This paper deals with the analogue of this conjecture over the field of formal Laurent series over $\mathbb{K}$, known as the $P(t)$-adic Littlew…
On the Hardy--Littlewood--Chowla conjecture
本文证明了哈代-李特尔伍德-乔拉猜想，作为应用得出在满足μ²(p+2)=1的素数中，μ(p+2)取1和-1的集合渐近密度均为1/2的结论。
On the Constants of the Lang-Trotter Conjecture for CM Elliptic Curves
本文通过检验 20 条 CM 椭圆曲线，验证 Wan–Xi 所得显式常数等于原 Lang-Trotter 常数，并确立相关猜想关于 r 的等价性。
Let $E$ be an elliptic curve over $\mathbb{Q}.$ Let $a_p$ denote the trace of the Frobenius endomorphism at a rational prime $p$. For a fixed integer $r,$ define the prime-counting function as $π_{E,r}(x):=\sum_{p\leq x,p\nmid Δ_E,a_p=r}1$. The Lang-Trotter Conjecture predicts that $$π_{E,r}(x)=C_{E,r}\cdot \frac{\sqrt{x}}{{\rm log}x}+o(\frac{\sqrt{x}}{{\rm log}x})$$ as $x\longrightarrow \infty,$ where $C_{E,r}$ is a specific non-negative constant. The Hardy-Littlewood Conjecture gives a similar asymptotic formula as above for the number of primes of the form $ax^2+bx+c$. We establish a relati…
Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teulié asked whether $\inf_{q \ge 1} \, q \cdot || q α|| \cdot | q |_p = 0$ holds for every badly approximable real number $α$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $α$ grows too rapidly or too slowly, then their conjecture is true for the pair $(α, p)$ with $p$ an arbitrary prime.
For any $s\in (1/2,1]$, the series$F_s(x)=\sum_{n=1}^{\infty} e^{iπn^2 x}/n^s$ converges almost everywhere on $[-1,1]$ by a result of Hardy-Littlewood, but not everywhere. However, there does not yet exist an intrinsic description of the set of convergence for $F_s$. In this paper, we define in terms of even or regular continued fractions certain subsets of points of $[-1,1]$ of full measure where the series converges. Our method is based on an approximate function equation for $F_s(x)$. As a by-product, we obtain the convergence of certain series defined in term of the convergents of the even…
An improvement of the lower bound of the number of integers in Littlewood's conjecture
本文改进了Littlewood猜想中满足不等式条件的整数数量的下界，证明在除低维集合外，该数量至少为\\(\gamma(\log N)^2/(\log\log N)^2\\)（至多差一个常数）。
In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares, which also explain the biases in the experimental data. In the process, we prove several …
Some Bounds Related to the $2$-adic Littlewood Conjecture
本文改进了$2$-adic Littlewood猜想反例的下界至15，并研究其B-变体证明界至少为5，同时利用Hurwitz算法发现具有特定等价性质的二次无理数族。
An elementary heuristic for Hardy-Littlewood extended Goldbach's conjecture
本文提出一个基于超几何分布和Hoeffding型界的初等组合启发式，用于预测Hardy-Littlewood扩展哥德巴赫猜想。
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