ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 4, pp. 398–402.
Pleiades Publishing, Inc., 2011.
Original Russian Text
G.A. Margulis, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 4, pp. 104–108.
THE INTERNATIONAL DOBRUSHIN PRIZE
Random Minkowski Theorem
G. A. Margulis
The International Dobrushin Prize for 2011 was awarded to Grigorii Aleksandrovich Margulis.
Margulis is aﬃliated with the Kharkevich Institute for Information Transmission Problems, Moscow,
and Yale University, New Haven, USA. The prize was presented on July 25, 2011, at the Interna-
tional Mathematical Conference in honor of the 50th anniversary of the Kharkevich Institute for
Information Transmission Problems of the Russian Academy of Sciences.
Grigorii Margulis’ Lecture
Roland L’vovich Dobrushin was my boss for more than twenty years. When I came to IPPI, I was
interested in pure mathematics only. Dobrushin pushed me, softly but persistently, in the direction
of applications. Somehow I resisted this pressure but nevertheless made some contributions to
coding theory and to what is now called theoretical computer science. One of these contributions was
explicit constructions of expander graphs, graphs with large girths, and low density codes. Another
contribution is about probabilistic connectivity of highly connected graphs. I should mention that
in the latter work Dobrishin helped me very much, pointing out to me a method which led to a
proof of one of the crucial statements.
Last time I talked to Dobrushin on the phone was in summer of 1995 when I visited Moscow.
At that time I did not know that he was gravely ill.
Roland L’vovich Dobrishin was a great scholar with broad interests both in mathematics and
outside of mathematics. He was also a remarkable person, and the warmest memories of him will
always stay with me.
In a joint paper  with J. Athreya, we obtained some results about logarithm laws for unipotent
ﬂows on the space of lattices SL(n, R)/ SL(n, Z). Our main tool there is a “random” analog of
Minkowski’s theorem on lattice points in convex bodies. Recall that Minkowski’s theorem states
that if A ⊂ R
is a convex, centrally symmetric region with m(A) > 2
(m is the Lebesgue measure
), then for all Λ ∈ X
there is a nonzero vector v ∈ Λ ∩ A,whereX
=SL(n, R)/ SL(n, Z)
denotes the space of unimodular lattices in R
Without the strong assumptions on the geometry of A, the result fails. However, one can ask
a probabilistic question: given a set A of large measure, what is the probability that a random
lattice Λ (chosen according to the Haar measure μ on X
) does not intersect A?
Theorem 1 [1, Theorem 2.2]. Let n ≥ 2. There is a constant C
such that if A is a measurable
set in R
with m(A) > 0 (m is the Lebesgue measure on R
μ(Λ ∈ X
:Λ∩ A = ∅) ≤
That is, the measure of the set of lattices that “miss” a set A is bounded above by a quantity inversely
proportional to the volume of A.