Sums of Reciprocals of Fractional Parts and Aspects of Multiplicative Diophantine Approximation

Reynold Fregoli

Research output: ThesisDoctoral Thesis

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Motivated by a question of Lê and Vaaler, we study upper bounds and asymptotic estimates for
sums of reciprocals of fractional parts in the multi-dimensional setting. These sums often occur
in Diophantine approximation, and a better understanding of their growth is of key importance
in multiple instances.
We obtain upper bounds by reducing the problem to counting lattice points in certain subsets
of R^n. These subsets are not easily treated because of their highly distorted shape and the
presence of "hyperbolic spikes". We develop an older partitioning method of Widmer to prove
a sophisticated counting result for weakly admissible lattices and sets with "hyperbolic spikes".
This counting result is the core of our thesis.
To apply our lattice-point counting result in the context of sums of reciprocals, we need to
introduce weaker types of multiplicative bad approximability for matrices, where the decay of
the product of fractional parts is controlled by a non-increasing function. We use this function
to make our bounds for sums of reciprocals explicit.
This raises questions on the existence of matrices with a prescribed type of multiplicative bad
approximability (in terms of the decay of the non-increasing function). In fact, it is well-known
that classical mutiplicatively badly approximable matrices are unlikely to exist, in connection
with the Littlewood conjecture. We prove the existence of matrices with a logarithmically
decaying type of bad approximability, by using an adaptation of a Cantor-type set construction
introduced by Badziahin and Velani. Matrices of this type have very low sums of reciprocals in
view of our previous estimate, and hence, provide a partial answer to Lê and Vaaler’s question.
Our bounds for sums of reciprocals can also be used to produce new examples of Khintchine
and strong Khintchine type vector subspaces of the Euclidean space, thus improving on certain
results of Huang and Liu.
The final chapter of this thesis is devoted to the solution of a problem posed by Lambert A’
Campo on the boundedness of certain exponential sums. We use a theorem of Duffin and Schaeffer
to settle the problem and we further investigate conditions under which such boundedness
is achieved. This involves the use of Cantor-type set constructions somewhat similar to those of
Badziahin and Velani.
Original languageEnglish
Awarding Institution
  • Royal Holloway, University of London
  • Widmer, Martin, Supervisor
  • Dietmann, Rainer, Advisor
Award date1 Nov 2020
Publication statusUnpublished - 2020


  • Number Theory
  • Diophantine Approximation
  • Sums of Reciprocals
  • Weak Admissibility
  • Exponential Sums

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