Motivation from continued fraction expansion:
Let us recall first the definition of continued fraction expansion (c.f.e).
Given a (possibly finite) sequence , we define
to be the limit of the convergents (this limit definition is analogous to presentation as decimal expansion).
It is not hard to show that for any sequence the induced sequence of convergents actually converge, and moreover every irrational number has a unique such presentation, and rational numbers have 2 presentations (e.g.
). While given any
it is easy to find
such that
, the importance of the convergents in the c.f.e is that they satisfy the inequality
. In other words, the c.f.e provides us with very good approximations to
.
The standard way to study these sort of presentations is using the shift left operator on the sequence , namely after we found the
-th convergent, we shift left
times in order to get the next coefficient
needed for the next convergent.
Definition (Gauss map): Given we define the Gauss map
to be
Note that for we have that
, hence the information about the
-th convergent is in the partial forward orbit
. It is well known that this orbit equidistribute for almost every
and it is a special instance of the pointwise ergodic theorem.
Pointwise Ergodic Theorem (for the Gauss map): For any and for almost every
we have that
Or equivalently we have the weak star limit
A point which satisfies the convergence above is called a generic point (for the Gauss map). It can be easily shown by using indicator functions, that a point is generic if and only if its coefficients in the continued fraction expansion satisfy a certain statistics (called the Gauss-Kuzmin statistics) and in particular every natural number should appear in its c.f.e. Thus, it is easy to construct points which are not generic and in particular we have the following three sets:
- The sequence
is bounded – corresponds to badly approximable numbers , namely
for which there exists some constant
for which
for any
.
- The sequence
is eventually periodic – corresponds to real algebraic numbers of degree 2.
- The sequence
is finite (in which case we cannot even take
to infinity in the PET) – corresponds to rational numbers.
By the PET, each of these sets is of zero Lebesgue (and Gauss) measure. The set of badly approximable numbers is very big in the sense that it is uncountable, has maximal Hausdorff dimension and is even Schmidt winning (so for example countably many intersection of its translations is still nonempty and has all the properties as above). On the other hand, the sets in (2) and (3) are countable. For these two sets, instead of taking a single element and an increasing sequence of its forward orbit, we can instead ask whether a suitable ordering of the elements will produce an equidistribution result. In this work we consider the case of the finite c.f.e.
Continued fraction expansion of rational numbers:
Let be a rational number where
are coprime. If
, then it easily seen that
where
and in particular
. Similarly we have that
where
and
Thus, the coefficient in the c.f.e of a rational number are exactly the quotients appearing in the Euclidean division algorithm. For example
Thus, trying to adapt the PET to these finite c.f.e we let be the length of the c.f.e of
(which is the number of steps in the Euclidean division algorithms) and then set
The first natural question is if for every choice of prime to
we have that
. This is false, since for example
is always a Dirac measure, and these measures cannot converge to
. On the other hand, this is a very specific example, and we might expect that there are very few such bad examples. This leads us to consider averages of the measures
over
(and
) and ask whether
We note that the normalization for each above is different according to
. For example, in the
case we normalize by the lengths
. If we want to use a uniform normalization we can ask instead whether
In this work we prove the convergence of the last two averages, and show that this implies that for almost every choice of
. These equidistribution results correspond to similar questions about equidistribution of geodesics in the space of 2-dimensional lattices (and have generalization to high dimension), which is where we prove them.
Divergent orbits in the space of lattices
All the results appearing here are from [1] (dimension 2) and [2] (general dimension).
Let be the space of unimodular
-dimensional lattices. We will use the following notation
where is the zero matrix with 1 in the
location.
There is a well known connection between the c.f.e of and the
-orbit
. Under this connection we get that
diverges as
, and the coefficients of the c.f.e can be seen in the forward orbit (i.e.
). In particular, the rational numbers can be characterized as the
for which the forward orbit diverges as well. In this case we obtain that the map
is a proper map (the preimage of a compact set is compact). This leads us to the following definition:
Definition: Let . The orbit
is called divergent if the map
is proper.
Remark: This divergence property in dimension 2 corresponds to the fact that the c.f.e of rational numbers are always finite.
Any rational lattice always contains a vector on each of the main axes, hence whenever
is large enough, there exists some
for which
is very small, hence
contains a very small nonzero vector, or in other words
is near the cusp. Thus the orbit of a rational lattice is always divergent, and as in the 2-dimensional case the converse is also true.
As in the c.f.e world, we define a measure for each orbit of a rational lattice. Recall that has the
dimensional Lebesgue measure, and we denote by
its push forward to the orbit
, and we note that for a divergent orbit this is a well defined
-invariant locally finite measure. In the case of dimension 2, the corresponding measure to
will be
.
As the measures above are only locally finite and not probability measures, we need to consider the “right” normalization, so we define the following convergence:
Definition: Let be nonzero locally finite measures on
.
- We write
if for any
we have that
.
- We write
if there exist some constants
such that
. Equivalently, for any
with
we have that
as
.
Under this notation, we have the following result:
Theorem 1: Let and set
. Then
where
is the
-invariant probability measure on
.
The main steps of the proof are as follows: Suppose that for some subsequence of
and a locally finite measure
. Then
-invariance: Since the
are
-invariant, then so is
.
- No escape of mass: We show that
is not the zero measure. This condition can be translated to a Diophantine condition on the elements in
, and we show that this condition holds by using the fact that
equidistributes in
as
. This equidistribution result is obvious if
runs over the prime numbers (or more generally a product of at most
primes for a fixed
), but this actually still holds for any sequence
.
- Maximal entropy: Normalizing
we obtain an
-invariant probability measure. For such measures we can compute the entropy
with respect to some nontrivial element
. We then use the theorem which states that
with equality if and only if
. Thus to complete the proof we prove that
must have the maximal entropy.
By definition, if is a divergent orbit, then
as
. In particular, there is a “nice” region
such that most of the “interesting” life span of
is actually in
and outside this region the orbit diverges quickly to infinity (more specifically, lattices in this part of the orbit are exactly those which do not have short vectors on the axes). Given
this region can be chosen uniformly over the orbits corresponding to the elements in
, and we denote this region by
. With this notation we have the following result which says that there are very few orbits for which their interesting life span is uniformly bounded.
Theorem 2: Fix some compact set and let
. Then:
- For
we have that
for some
.
- For
we have that
for any
.
Finally, for dimension 2, the theorem above can be translated back to c.f.e, and asks for a fixed how many
coprime to
are there such that the coefficients in the c.f.e of
are bounded by
. Let us denote this set by
. A well known conjecture regarding this set is Zaremba’s conjecture which asks to show that there exists some constant
for which
for all
. What is known (see Bourgain and Kontorovich [3] and Huang [4]) is that for
, these sets are not empty for almost every
. The theorem above shows that even if this set is not empty, it cannot be too big. More specifically, we have the following:
Corollary 3: For any there exists some
such that
.
Upgrading the result:
Using the ergodicity of the uniform measure with respect to the group
and the fact that the measures
are averages of
-orbits, we can upgrade Theorem 2 as follows:
Upgrade 1: Let such that
. Then
.
This theorem follows from the presentation
By going to a subseqeunce on which all the averages above and their coefficients converge, we may assume that , hence taking the limit (with the proper normalization) we obtain that
.
It is easy to show that both and
must be
-invariant probability measure. The ergodicity of
implies that it is an extreme point in the set of such measure, i.e. it cannot be written as a proper convex combination of two
-invariant probability measure. From that we conclude that
which implies Upgrade 1.
Next, we upgrade this result even further and show that almost all the for
are close to the uniform measure (after the suitable normalization).
Upgrade 2: There exist sets such that
- We have that
, and
- For any choice of sequence
we have that
.
The main idea of the proof of this upgrade is that if it is not true, then after a suitable normalization we could find a positive proportion of the elements in and some witness function
for which
are “far” from
. But such a result will contradict the previous upgrade.
In particular, for dimension 2 and its connection to c.f.e, we obtain the following:
Corollary: There exist sets such that
- We have that
, and
- For any choice of sequence
we have that
.
Divergent orbits in the space of adelic lattices
In the results in the previous section in the space we took an average over some set
and then an average over the diagonal flow. The set
always consisted of rational lattices with vectors defined over
. Of course, there are many more such lattices than just those inside
and a natural question is what makes this set so important, and are there any other nice sets which have similar equidistribution results. In order to see this result in a much more natural way we lift the discussion to the space of adelic lattices where the average over the rationals can be see as a translation in the p-adic places.
Recall first that we have a natural projection
defined as follows. Given where
and
such that in almost every place
, we can always find
such that
for all
. We then define the projection
.
Letting be the point corresponding to
and
be the set of diagonal matrices in
we denote by
the locally finite
-invariant measure on the orbit
. It is not hard to check that the projection of this measure to
is exactly
, namely the
-invariant measure on the orbit
. When the measure
is first translated by an element from
which is trivial in the real place and then projected down to
, then this translation turns into an average over several orbit measures. In particular we are interested in translation by elements of the form
One can show that for we have that
, and we use this fact to prove that:
Theorem 4: The sequence converges to the
-invariant measure on
.
This theorem leads to the more general question of finding a condition of a sequence for which the translations
equidistribute. The theorem above shows that we can choose the sequence
.
Note that the sequence cannot equidistribute if are in some compact set. Moreover, since
is
-invariant, then a necessary condition is that
diverges modulo
. On the other hand, if the sequence
equidistributes, then clearly
equidistributes for any choice of
and
in a fixed compact set. Thus, using the Iwasawa decomposition in the
we obtain the following:
Theorem 5: Let which satisfy:
- The sequence diverges modulo
, and
- The real part of the
is trivial for each
(or more generally, it is in a compact set modulo
).
Then the sequence equidistributes.
Bibliography:
- O. David, U. Shapira, “Equidistribution of divergent orbits and continued fraction expansion of rationals”, arXiv: arXiv:1707.00427 [math.DS]
- O. David, U. Shapira, “Equidistribution of divergent orbits of the diagonal group in the space of lattices”, arXiv: arXiv:1710.05242 [math.DS]
- J. Bourgain and A. Kontorovich, “On Zaremba’s conjecture”. Annals of Mathematics, 180(1), pp.137-196, 2014.
- S. Huang, “An improvement to Zaremba’s conjecture”. Geometric and Functional Analysis, 25(3), pp.860-914, 2015.