Continued fractions, Bohr sets, and the Littlewood conjecture

Let be an element of the unit circle, let , and let . We define the (rank one) Bohr set to be the set

where is the distance to the origin in the unit circle (or equivalently, the distance to the nearest integer, after lifting up to ). These sets play an important role in additive combinatorics and in additive number theory. For instance, they arise naturally when applying the circle method, because Bohr sets describe the oscillation of exponential phases such as .

Observe that Bohr sets enjoy the doubling property

thus doubling the Bohr set doubles both the length parameter and the radius parameter . As such, these Bohr sets resemble two-dimensional balls (or boxes). Indeed, one can view as the preimage of the two-dimensional box under the homomorphism .

Another class of finite set with two-dimensional behaviour is the class of (rank two) generalised arithmetic progressions

with and Indeed, we have

and so we see, as with the Bohr set, that doubling the generalised arithmetic progressions doubles the two defining parameters of that progression.

More generally, there is an analogy between rank Bohr sets

and the rank generalised arithmetic progressions

One of the aims of additive combinatorics is to formalise analogies such as the one given above. By using some arguments from the geometry of numbers, for instance, one can show that for any rank Bohr set , there is a rank generalised arithmetic progression for which one has the containments

for some explicit depending only on (in fact one can take ); this is (a slight modification of) Lemma 4.22 of my book with Van Vu.

In the special case when , one can make a significantly more detailed description of the link between rank one Bohr sets and rank two generalised arithmetic progressions, by using the classical theory of continued fractions, which among other things gives a fairly precise formula for the generators and lengths of the generalised arithmetic progression associated to a rank one Bohr set . While this connection is already implicit in the continued fraction literature (for instance, in the classic text of Hardy and Wright), I thought it would be a good exercise to work it out explicitly and write it up, which I will do below the fold.

It is unfortunate that the theory of continued fractions is restricted to the rank one setting (it relies very heavily on the total ordering of one-dimensional sets such as or ). A higher rank version of the theory could potentially help with questions such as the Littlewood conjecture, which remains open despite a substantial amount of effort and partial progress on the problem. At the end of this post I discuss how one can use the rank one theory to rephrase the Littlewood conjecture as a conjecture about a doubly indexed family of rank four progressions, which can be used to heuristically justify why this conjecture should be true, but does not otherwise seem to shed much light on the problem.

Distinguished Lecture Series I: Gregory Margulis, “Homogeneous dynamics and number theory I.”

The final distinguished lecture series for the academic year here at UCLA is being given this week by Gregory Margulis, who is giving three lectures on “homogeneous dynamics and number theory”.  In his first lecture, Prof. Margulis surveyed some classical problems in number theory that turn out, rather surprisingly, to have more or less equivalent counterparts in homogeneous dynamics – the theory of dynamical systems on homogeneous spaces .

As usual, any errors in this post are due to my transcription of the talk.