||Hill Center 323
||pcohen "at" bowdoin "dot" edu
||Local-Global Phenomena and Applications to Zaremba's Conjecture and Circle Packings
In many different areas of mathematics, areas ranging from numerical integration techniques to Apollonian gaskets, the same phenomena manifest themselves.
Namely, given a particular semi-group of square matrices with dimension d given by Γ acting on a vector
of ℤ d one cares to determine whether a given vector is barred from an orbit
under said semi-group only by congruence relations. This phenomena is known as the local-global phenomena.
For this project, we hope to answer questions related to such local-global phenomena in the context of
Zaremba's Conjecture as well as in Integral Apollonian Circle Packings.
From Apollonius To Zaremba:
Local-Global Phenomena In Thin Orbits by A. Kontorovich
Integral Apollonian Packings
by P. Sarnak
- Week 1:
- After selecting my project, I created a short presentation (to view the presentation, see the link below) to present my area and direction research.
After presenting, I continued to develop my knowledge of the requisite background information further.
- Week 2:
- Over the course of this week, I worked through the singular series for the Zaremba's Conjecture multiplicity problem following from the guidelines of
Shinnyih Huang's paper. To do this, I computed the size of SL(2, ℤ / q ℤ), analysed
Ramanujan sums averaged over the group SL(2, ℤ / q ℤ), and formed an Euler product for the singular series that I had developed last week.
After completing this, I began to simulate the growth of the subgroup, ΓA, of SL(2, ℤ) that is associated with Zaremba's Conjecture using C++.
- Week 3:
- I spent the week developing an efficient algorithm in C++ to analyse multiplicities in ΓA. To this end, I designed an algorithm that, in a
sense, recursively constructs the minimal amount of the Cayley graph of the ΓA needed to achieve the Archimedean ball of a given radius.
While this subsection of the Cayley graph is being built the program simultaneously determines the multiplicity of each value within the ball.
By implementing the program as such, very large Archimedean balls can be examined in a relatively short period of time (a matter of minutes on an ordinary desktop
computer). Thus far, all simulations have been conducted for A = 5, where A is the constant value associated to Zaremba's Conjecture. We now have multiplicity
data for all target values up to 254238.
- Week 4:
- This week, I spent my time testing the accuracy of our singular series against the simulated data developed last week. With the singular series, I
generated ratios of expected multiplicities for consecutive natural numbers by using our singular series and compared this data to the actual ratios of multiplicities for the same
consecutive numbers. The results of this demonstrate that the singular series does, in fact, asymptotically approximate the true multiplicities of natural numbers.
To help visualize this test, I have added two images below (Convergence Image 1 and Convergence Image 2).
- Week 5:
- At the start of the week, we developed a conjecture regarding the multiplicities of target values in ΓA. We spent the week justifying our
conjecture. The week culminated with a justification of our conjecture, which demonstrated that our singular series does indeed behave as it ought to. We may now
modify the projection mapping from ΓA to the natural numbers and attempt to develop a new singular series. Another possibility for future research
entails completing the major arc analysis of our original singular series, in order to conform to the standard methods of Hardy and Littlewood.
- Week 6:
- Over the course of this week, I worked on creating a heuristic, similar to the one that we had already developed, for computing multiplicities given the
trace projection rather than the projection that selects the bottom right-hand element. While the general set-up for this approach is similar, it revolves around the
same concepts of the Hardy-Littlewood circle method, new difficulties arose in actually evaluating the singular series. I spent the remainder of the week attempting
to generalize Huang's results for the bottom right-hand element projection mapping to the trace mapping. On Friday, I presented my research; I have added my
presentation below (Second Presentation).
- Week 7:
- I spent the week changing the simulation software to work around a compiler-based restriction on the size of target values. To this end, I now have data
for target values ranging from 1 to 900000. However, new system-based limitations have restricted my capability to work with this data. While waiting for the
requisite software to work effectively with the data generated, I also worked further on developing a singular series for the multiplicity question based upon the
trace projection. This singular series seems to be much less pretty than the singular series for the bottom right-hand element projection.