Sums of Reciprocals of Fractional Parts and Aspects of Multiplicative Diophantine Approximation. / Fregoli, Reynold.

2020. 112 p.

Research output: ThesisDoctoral Thesis

Unpublished

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@phdthesis{b771e86a775c41d3b6650810ae234c86,
title = "Sums of Reciprocals of Fractional Parts and Aspects of Multiplicative Diophantine Approximation",
abstract = "Motivated by a question of L{\^e} and Vaaler, we study upper bounds and asymptotic estimates forsums of reciprocals of fractional parts in the multi-dimensional setting. These sums often occurin Diophantine approximation, and a better understanding of their growth is of key importancein multiple instances.We obtain upper bounds by reducing the problem to counting lattice points in certain subsetsof R^n. These subsets are not easily treated because of their highly distorted shape and thepresence of {"}hyperbolic spikes{"}. We develop an older partitioning method of Widmer to provea 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 tointroduce weaker types of multiplicative bad approximability for matrices, where the decay ofthe product of fractional parts is controlled by a non-increasing function. We use this functionto make our bounds for sums of reciprocals explicit.This raises questions on the existence of matrices with a prescribed type of multiplicative badapproximability (in terms of the decay of the non-increasing function). In fact, it is well-knownthat classical mutiplicatively badly approximable matrices are unlikely to exist, in connectionwith the Littlewood conjecture. We prove the existence of matrices with a logarithmicallydecaying type of bad approximability, by using an adaptation of a Cantor-type set constructionintroduced by Badziahin and Velani. Matrices of this type have very low sums of reciprocals inview of our previous estimate, and hence, provide a partial answer to L{\^e} and Vaaler{\textquoteright}s question.Our bounds for sums of reciprocals can also be used to produce new examples of Khintchineand strong Khintchine type vector subspaces of the Euclidean space, thus improving on certainresults of Huang and Liu.The final chapter of this thesis is devoted to the solution of a problem posed by Lambert A{\textquoteright}Campo on the boundedness of certain exponential sums. We use a theorem of Duffin and Schaefferto settle the problem and we further investigate conditions under which such boundednessis achieved. This involves the use of Cantor-type set constructions somewhat similar to those ofBadziahin and Velani.",
keywords = "Number Theory, Diophantine Approximation, Sums of Reciprocals, Weak Admissibility, Exponential Sums",
author = "Reynold Fregoli",
year = "2020",
language = "English",
school = "Royal Holloway, University of London",

}

RIS

TY - THES

T1 - Sums of Reciprocals of Fractional Parts and Aspects of Multiplicative Diophantine Approximation

AU - Fregoli, Reynold

PY - 2020

Y1 - 2020

N2 - Motivated by a question of Lê and Vaaler, we study upper bounds and asymptotic estimates forsums of reciprocals of fractional parts in the multi-dimensional setting. These sums often occurin Diophantine approximation, and a better understanding of their growth is of key importancein multiple instances.We obtain upper bounds by reducing the problem to counting lattice points in certain subsetsof R^n. These subsets are not easily treated because of their highly distorted shape and thepresence of "hyperbolic spikes". We develop an older partitioning method of Widmer to provea 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 tointroduce weaker types of multiplicative bad approximability for matrices, where the decay ofthe product of fractional parts is controlled by a non-increasing function. We use this functionto make our bounds for sums of reciprocals explicit.This raises questions on the existence of matrices with a prescribed type of multiplicative badapproximability (in terms of the decay of the non-increasing function). In fact, it is well-knownthat classical mutiplicatively badly approximable matrices are unlikely to exist, in connectionwith the Littlewood conjecture. We prove the existence of matrices with a logarithmicallydecaying type of bad approximability, by using an adaptation of a Cantor-type set constructionintroduced by Badziahin and Velani. Matrices of this type have very low sums of reciprocals inview 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 Khintchineand strong Khintchine type vector subspaces of the Euclidean space, thus improving on certainresults 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 Schaefferto settle the problem and we further investigate conditions under which such boundednessis achieved. This involves the use of Cantor-type set constructions somewhat similar to those ofBadziahin and Velani.

AB - Motivated by a question of Lê and Vaaler, we study upper bounds and asymptotic estimates forsums of reciprocals of fractional parts in the multi-dimensional setting. These sums often occurin Diophantine approximation, and a better understanding of their growth is of key importancein multiple instances.We obtain upper bounds by reducing the problem to counting lattice points in certain subsetsof R^n. These subsets are not easily treated because of their highly distorted shape and thepresence of "hyperbolic spikes". We develop an older partitioning method of Widmer to provea 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 tointroduce weaker types of multiplicative bad approximability for matrices, where the decay ofthe product of fractional parts is controlled by a non-increasing function. We use this functionto make our bounds for sums of reciprocals explicit.This raises questions on the existence of matrices with a prescribed type of multiplicative badapproximability (in terms of the decay of the non-increasing function). In fact, it is well-knownthat classical mutiplicatively badly approximable matrices are unlikely to exist, in connectionwith the Littlewood conjecture. We prove the existence of matrices with a logarithmicallydecaying type of bad approximability, by using an adaptation of a Cantor-type set constructionintroduced by Badziahin and Velani. Matrices of this type have very low sums of reciprocals inview 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 Khintchineand strong Khintchine type vector subspaces of the Euclidean space, thus improving on certainresults 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 Schaefferto settle the problem and we further investigate conditions under which such boundednessis achieved. This involves the use of Cantor-type set constructions somewhat similar to those ofBadziahin and Velani.

KW - Number Theory

KW - Diophantine Approximation

KW - Sums of Reciprocals

KW - Weak Admissibility

KW - Exponential Sums

M3 - Doctoral Thesis

ER -