Manfred Einsiedler, Dmitry Kleinbock
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.
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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.
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.
The essential aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of p-adic q-invariant distribution on Zp. Moreover, we give some interesting properties concerning this type maximal operator.
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.
Positive Logarithmic Hausdorff Measures of Exceptional Sets for the $p$-adic and $t$-adic Littlewood Conjectures
该研究证明p进Littlewood猜想例外集若非空则具正对数Hausdorff测度，还得到有限域上t进情形的更强结论，改进了Lai和Sprang的构造，特征2情形的结论仍待反例存在性验证。
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。
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 fundamental aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of fermionic p-adic q-invariant distribution on Zp . Also, we derive some interesting properties concerning this type maximal operator.
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…
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.
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。
Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors
本文研究了代数向量的位移大小与方向分布，给出了Bugeaud关于代数向量满足p-adic Littlewood猜想结果的新证明，并利用该证明对逼近缺陷方向序列的极限分布进行了分类。
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 prove that the boundary of the Hall-Littlewood $t$-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin and Cuenca on boundaries of related deformed Gelfand-Tsetlin graphs. In the special case when $1/t$ is a prime $p$ we use this to recover results of Bufetov-Qiu and Assiotis on infinite $p$-adic random matrices, placing them in the general context of branching graphs derived from symmetric functions. Our methods rely on explicit formulas for certain skew Hall-Littlewood polynomials. As a separate corollary to these, we obtain …
The $p$-adic Littlewood Conjecture due to De Mathan and Teulié asserts that for any prime number $p$ and any real number $α$, the equation $$\inf_{|m|\ge 1} |m|\cdot |m|_p\cdot |\langle mα\rangle|\, =\, 0 $$ holds. Here, $|m|$ is the usual absolute value of the integer $m$, $|m|_p$ its $p$-adic absolute value and $ |\langle x\rangle|$ denotes the distance from a real number $x$ to the set of integers. This still open conjecture stands as a variant of the well-known Littlewood Conjecture. In the same way as the latter, it admits a natural counterpart over the field of formal Laurent series $\ma…
We extend the work of Ash and Stevens [Ash-Stevens 97] on p-adic analytic families of p-ordinary arithmetic cohomology classes for GL(N,Q) by introducing and investigating the concept of p-adic rigidity of arithmetic Hecke eigenclasses. An arithmetic eigenclass is said to be "rigid" if (modulo twisting) it does not admit a nontrivial p-adic deformation containing a Zariski dense set of arithmetic specializations. This paper develops tools for explicit investigation into the structure of eigenvarieties for GL(N). We use these tools to prove that known examples of non-sefldual cohomological cusp…
Rigidity and unlikely intersections for stable $p$-adic dynamical systems
针对 Berger 关于稳定 $p$-进幂级数预周期点能否确定稳定 $p$-进动力系统的问题，本文利用这些预周期点实现了对相应动力系统的确定。
On the $P(t)$-adic Littlewood Conjecture in Characteristics $\ell \equiv 3\pmod{4}$
本研究证明特征满足$\ell \equiv 3\pmod{4}$时$P(t)$进Littlewood猜想不成立，通过揭示-1非二次剩余导致的算术障碍，推广了此前仅能验证小特征的计算机辅助方法，为剩余奇特征情形的研究提供了显式方向。
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。
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…
$p$-adic rigidity for $\mathrm{GSp}_4$
本文研究了$\mathrm{GSp}_4$上特定非 cuspidal 自动形 Saito-Kurokawa 表示的$p$-adic 刚性，揭示了其在$p$-adic 家族中无法非平凡插值的性质，并识别了$\mathrm{GSp}_4$特征谱上的$p$-adic 变化障碍。
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.
We construct a functor from the category of p-adic etale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection over its "base change to B_dR", which can be regarded as a first step towards the sought-after p-adic Riemann-Hilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at ev…
Motivated by questions about $\mathbb{C}_p$-valued Fourier transform on the locally compact group $(\mathbb{Q}_p^d,+)$, we study invariant norms on the $p$-adic Schrödinger representation of the Heisenberg group. Our main result is a minimality and rigidity property for norms in a family of invariant norms parameterized by a Grassmannian. This family is the orbit of the sup norm under the action of the symplectic group, acting via intertwining operators. We also prove general fundamental properties of quotients of the universal unitary completion of cyclic algebraic representations. Combined w…
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。
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…
$p$-adic rigidity of eigenforms of infinite slope
本文提出了允许斜率为无穷的 Hecke 特征形 $p$-进族概念，并证明了无穷斜率族仅存在特定扭曲或复乘形式，其证明结合了局部 Galois 表示形变与全局模形式猜想。
Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration
本文引入Hall-Littlewood-Schubert级数，通过变量替换解决p-adic格枚举等问题，提供Hecke级数新公式，并作为Stanley-Reisner环Hilbert级数的q-模拟，推广Littlewood恒等式。
Monodromy and rigidity of crystalline local systems
本文研究p-进域上光滑刚性解析空间X上p-进局部系统的刚性性质，证明对数晶状晶体单值性沿特殊纤维不可约分支的刚性，并给出多方面应用，包括Shankar猜想的证明及局部系统的唯一延拓性等。
Let K be a p-adic field and let F and G be two formal groups over O_K. We prove that if F and G have infinitely many torsion points in common, then F=G. This follows from a rigidity result: any bounded power series that sends infinitely many torsion points of F to torsion points of F is an endomorphism of F.
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。
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。
p-adic Rigidity and Supercongruences at Rank Two Attractors
本文研究秩二吸引子引发的p-adic刚性，提出上同调判据证明Frobenius不动点与超同余，并应用于Hulek--Verrill A_4族及镜像四次K3铅笔。
We prove $p$-adic versions of a classical result in arithmetic geometry stating that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a $p$-adic field $K$ or its ring of integers $R$, respectively. In particular, we show that the rigidity results for algebraic functions underlying the so-called Manin-Mumford Conjecture generalize to suitable $p$-adic analytic functions. In the formal setting, this approach leads us to uncover purely $p$-adi…
Our results concern analytic functions on the open unit $p$-adic poly-disc in $\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $(ζ_1-1,\ldots,ζ_n-1)$ unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin-Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond just …
Measure rigidity is a branch of ergodic theory that has recently contributed to the solution of some fundamental problems in number theory and mathematical physics. Examples are proofs of quantitative versions of the Oppenheim conjecture, related questions on the spacings between the values of quadratic forms, a proof of quantum unique ergodicity for certain classes of hyperbolic surfaces, and an approach to the Littlewood conjecture on the nonexistence of multiplicatively badly approximable numbers. In these lectures we discuss a few simple applications of one of the central results in measur…
Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-Kähler varieties
本文证明了超Kähler流形在特定条件下的Mumford-Tate猜想，并展示了其在变形下的不变性，同时推导出Hodge猜想与Tate猜想的关系。
On the ordinary Hecke orbit conjecture
该研究证明了良好约化素数下霍奇型志村簇的常赫克轨道猜想，借助Chai的Serre-Tate坐标与D'Addezio的p进单值群结果，提出新的单值群定理与刚性结果，还推广了Howe关于Serre-Tate坐标的结论。
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