American Institute of Mathematical Sciences

Advanced
July  2009, 3(3): 457-477. doi: 10.3934/jmd.2009.3.457

On a generalization of Littlewood's conjecture

Uri Shapira 1,

1. 

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

Received  April 2009 Revised  June 2009 Published  August 2009

We present a class of lattices in $\R^d$ ($d\ge 2$) which we call grid-Littlewood 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 grid-Littlewood. We then prove the existence of grid-Littlewood lattices by first establishing a dimension bound for the set of possible exceptions. The existence of vectors (grid-Littlewood-vectors) in $\R^d$ with special Diophantine properties is proved by similar methods. Applications to Diophantine approximations are given. For dimension $d\ge 3$, we give explicit constructions of grid-Littlewood lattices (and in fact lattices satisfying a much stronger property). We also show that GLC is implied by a conjecture of G. A. Margulis concerning bounded orbits of the diagonal group. The unifying theme of the methods is to exploit rigidity results in dynamics ([4, 1, 5]), and derive results in Diophantine approximations or the geometry of numbers.
Keywords: diagonal flow, Littlewood's conjecture, inhomogeneous lattices, product of inhomogeneous linear forms..
Mathematics Subject Classification: Primary: 37A17, 37N99; Secondary: 11H46, 11J20, 11J25, 11J3.
Citation: Uri Shapira. On a generalization of Littlewood's conjecture. Journal of Modern Dynamics, 2009, 3 (3) : 457-477. doi: 10.3934/jmd.2009.3.457
[1]

Jiyoung Han. Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2205-2225. doi: 10.3934/dcds.2020359
[2]

Tong Li. Well-posedness theory of an inhomogeneous traffic flow model. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 401-414. doi: 10.3934/dcdsb.2002.2.401
[3]

Hiroko Morimoto. Survey on time periodic problem for fluid flow under inhomogeneous boundary condition. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 631-639. doi: 10.3934/dcdss.2012.5.631
[4]

Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143
[5]

Raimund Bürger, Antonio García, Kenneth H. Karlsen, John D. Towers. Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks & Heterogeneous Media, 2008, 3 (1) : 1-41. doi: 10.3934/nhm.2008.3.1
[6]

JinMyong An, JinMyong Kim, KyuSong Chae. Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021221
[7]

D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843
[8]

Ionuţ Munteanu. Boundary stabilization of non-diagonal systems by proportional feedback forms. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3113-3128. doi: 10.3934/cpaa.2021098
[9]

Fioralba Cakoni, Anne Cossonnière, Houssem Haddar. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Problems & Imaging, 2012, 6 (3) : 373-398. doi: 10.3934/ipi.2012.6.373
[10]

Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477
[11]

Fang Zeng, Xiaodong Liu, Jiguang Sun, Liwei Xu. The reciprocity gap method for a cavity in an inhomogeneous medium. Inverse Problems & Imaging, 2016, 10 (3) : 855-868. doi: 10.3934/ipi.2016024
[12]

Fenglong Qu, Jiaqing Yang. On recovery of an inhomogeneous cavity in inverse acoustic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 281-291. doi: 10.3934/ipi.2018012
[13]

Miroslav Bulíček, Eduard Feireisl, Josef Málek, Roman Shvydkoy. On the motion of incompressible inhomogeneous Euler-Korteweg fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 497-515. doi: 10.3934/dcdss.2010.3.497
[14]

Ola I. H. Maehlen. Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4113-4130. doi: 10.3934/dcds.2020174
[15]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337
[16]

Mohammad Asadzadeh, Anders Brahme, Jiping Xin. Galerkin methods for primary ion transport in inhomogeneous media. Kinetic & Related Models, 2010, 3 (3) : 373-394. doi: 10.3934/krm.2010.3.373
[17]

Carmen Cortázar, Manuel Elgueta, Jorge García-Melián, Salomé Martínez. Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1409-1419. doi: 10.3934/dcds.2015.35.1409
[18]

Michael P. Mortell, Brian R. Seymour. Resonant oscillations of an inhomogeneous gas in a closed cylindrical tube. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 619-628. doi: 10.3934/dcdsb.2007.7.619
[19]

Liping Wang, Juncheng Wei. Infinite multiplicity for an inhomogeneous supercritical problem in entire space. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1243-1257. doi: 10.3934/cpaa.2013.12.1243
[20]

Yakov Varshavsky. A proof of a generalization of Deligne's conjecture. Electronic Research Announcements, 2005, 11: 78-88.

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences