zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Invariant measures and the set of exceptions to Littlewood’s conjecture. (English) Zbl 1109.22004

There is a well-known and long-standing conjecture of Littlewood: u,v,

lim inf n nnunv=0,

where w=min n |w-n| is the distance of w to the nearest integer. Let A be the group of positive diagonal k×k matrices on SL(k,)/SL(k,). In the paper under review some results which have implications on Littlewood’s conjecture are proven. Main results of the paper are:

1) Let μ be an A-invariant and ergodic measure on X=SL(k,)/SL(k,) for a subgroup of A which acts on X with positive entropy. Then μ is algebraic.

2) Let Ξ={(u,v) 2 :lim inf n nnunv>0}. Then the Hausdorff dimension

dim H Ξ=0·

3) For any k linear forms m i (x 1 ,,x k )= j=1 k m ij (x j ) and f m (x 1 ,,x k )= i=1 k m i (x 1 ,,x k ), where m=(m ij ) denotes the k×k matrix whose rows are the linear forms m i (x 1 ,,x k ), there is a set Ξ k SL(k,) of Hausdorff dimension k-1 so that mSL(k,)Ξ k ,

inf x k {0} |f m (x)|=0·

The last result has applications to a generalization of Littlewood’s conjecture.


MSC:
22F10Measurable group actions
11J13Simultaneous homogeneous approximation, linear forms
37A45Relations of ergodic theory with number theory and harmonic analysis
28A80Fractals
37A15General groups of measure-preserving transformation
11H46Products of linear forms