INTERSECTIONS OF DELETED DIGITS CANTOR SETS WITH GAUSSIAN INTEGER BASES Athesissubmittedinpartialfulfillment of the requirements for the degree of Master of Science By VINCENT T. SHAW B.S., Wright State University, 2017 2020 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES May 1, 2020 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPER- VISION BY Vincent T. Shaw ENTITLED Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases BE ACCEPTED IN PARTIAL FULFILLMENT OF THE RE- QUIREMENTS FOR THE DEGREE OF Master of Science. ____________________ Steen Pedersen, Ph.D. Thesis Director ____________________ Ayse Sahin, Ph.D. Department Chair Committee on Final Examination ____________________ Steen Pedersen, Ph.D. ____________________ Qingbo Huang, Ph.D. ____________________ Anthony Evans, Ph.D. ____________________ Barry Milligan, Ph.D. Interim Dean, School of Graduate Studies Abstract Shaw, Vincent T. M.S., Department of Mathematics and Statistics, Wright State University, 2020. Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases. In this paper, the intersections of deleted digits Cantor sets and their fractal dimensions were analyzed. Previously, it had been shown that for any dimension between 0 and the dimension of the given deleted digits Cantor set of the real number line, a translate of the set could be constructed such that the intersection of the set with the translate would have this dimension. Here, we consider deleted digits Cantor sets of the complex plane with Gaussian integer bases and show that the result still holds. iii Contents 1 Introduction 1 1.1 WhystudyintersectionsofCantorsets? . 1 1.2 Prior work on this problem . 1 2 Negative Base Representations 5 2.1 IntegerRepresentations .............................. 5 2.2 Fractals Obtained by Deleting Digits . 7 3 Gaussian Integers and Fractals 9 3.1 GaussianIntegerRepresentations . 9 3.2 Gaussian Fractals Obtained by Deleting Digits . 13 4 Preliminaries 15 4.1 Entropy Dimension of the Intersections . 15 4.2 Construction of Deleted Digits Cantor Sets . 17 4.3 Investigating T (T + x) ............................. 20 \ 5 Proof of Theorem 21 6 Figures and Tables 23 7 Conclusion 32 iv List of Figures 6.1 T , b = 2+i .................................... 23 0 − 6.2 T , b = 2+i, D⇤ = 0, 3 ............................ 24 1 − { } 6.3 T , b = 2+i, D⇤ = 0, 3 ............................ 25 2 − { } 6.4 T (T + 3) = , b = 2+i, D⇤ = 0, 3 .................... 26 0 \ 0 ; − { } 6.5 T (T +0.3) ................................... 27 1 \ 1 6.6 T (T +0.30) ................................... 28 2 \ 2 6.7 T (T +0.33) ................................... 29 2 \ 2 v List of Tables 6.1 b = 2+i, D⇤ = 0, 3 .............................. 30 − { } 6.2 b = 3+i, D⇤ = 0, 6 .............................. 30 − { } 6.3 b = 4+i, D⇤ = 0, 8 .............................. 31 − { } vi Acknowledgements First and foremost, I would like to thank my research advisor Dr. Steen Pedersen for providing me the opportunity to study mathematics under his guidance and leadership. The past two years working with him have given me a new found appreciation for those that contribute to field of mathematics. It was only through his encouragement, and profound patience that this thesis was possible. Also, special thanks is in order for Jason D. Phillips for his coauthorship to the predecessor of this paper that laid the foundation for our research. Thank you to both Dr. Qingbo Huang and Dr. Anthony Evans for reviewing my thesis and providing feedback. I would also like to take a moment to thank the following individuals that had a positive impact on my personal development as a scholar: Dr. Ayse Sahin, Dr. Ann Farrell and Peggy Kelly in helping me succeed as a Graduate Teaching Assistant. Thank you to both Dr. Christopher Barton and Dr. Mateen Rizki for taking me on as a student in their MatLab course that proved to be an invaluable resource in conducting my research. Finally, I would like to thank my friend and colleague Joe Sjoberg for rekindling my interest in pursuing a degree in mathematics, as well as my fellow GTA’s for making the last two years an enjoyable and worthwhile expreience. vii viii Chapter 1 Introduction 1.1 Why study intersections of Cantor sets? Cantor sets may appear to be rather special. However, they occur in mathematical models of many naturally occuring objects, play a role in number theory, in signal processing, in er- godic theory, and in limit-theorems from probability. We study the “size” of the intersection of two Cantor sets. The significance of this problem was noted by Furstenberg [Fur70] and Palis [Pal87]. Papers dealing with versions of our problem and related applications include: [Wil91], [Kra92], [PS98], [Kra00], [LN04], [DT07], [DHW08], [DT08a], [ZLL08], [DLT09], [KLD10], [LYZ11a], [Mor11], [ZG11], [LTZ18], [SRM+20]. The literature in the subject, neighboring areas, and applications is vast. In the list above, we limit ourselves to a small sample of the literature closely related to our problem. 1.2 Prior work on this problem Most of the prior work related to our problem has been done in the one dimensional case, so we will begin with a summary of that case. 1 Let n 3 be an integer. Any real number t [0, 1] has at least one n-ary representation ≥ 2 1 t t =0. t t = k n 1 2 ··· nk kX=1 where each t is one of the digits 0, 1,...,n 1. Deleting some element from the full digit k − set 0, 1,...n 1 we get a set of digits D := d k =1, 2,...,m with m<ndigits { − } { k | } dk <dk+1 and a corresponding deleted digits Cantor set 1 xk C = Cn,D := k xk D for all k N . (1.1) ( n | 2 2 ) kX=1 We are interested in the dimension of the sets C (C + t) , where C + t := x + t x C . \ { | 2 } Since the problem is invariant under translation we will assume d1 =0. We say that D is uniform, if d d ,k=1, 2,...,m 1 is constant and 2. We k+1 − k − ≥ say D is regular, if D is a subset of a uniform digit set. Finally, we say that D is sparse, if δ δ0 2 for all δ = δ0 in | − | ≥ 6 ∆ := D D = d d d ,d D . − { j − k | j k 2 } Clearly, a uniform set is regular and a regular set is sparse. The set D = 0, 5, 7 is sparse { } and not regular. We abuse the terminology and say Cn,D is uniform, regular, or sparse provided D has the corresponding property. Previous studies of the sets C (C + t) include: \ When C = C3, 0,2 is the middle thirds Cantor set, a formula for the Hausdorff • { } dimension dim (C (C + t)) of C (C + t) can be found in [DH95] and in [NL02]. \ \ Such a formula can also be found in [DT08b] if C is uniform and d = n 1,andin m − [KP91] if C is regular. For sparse C, aformulacanbefoundin[PP13]. Let F + be the set of all t 0 such that C (C + t) is non-empty. For 0 β 1, let • ≥ \ F := t F + dim (C (C + t)) = β log (m) . If C is the middle thirds Cantor set β { 2 | \ n } + + then F =[0, 1] and it is shown in [Haw75, DH95, NL02] that Fβ is dense in F for 2 all 0 β 1. This is extended to regular sets and to sets C such that D satisfies n,D d d 2 and d <n 1 in [PP12]. It is also shown in [PP12] that F is not k+1 − k ≥ m − β dense in F + for all 0 β 1 for all deleted digits Cantor sets C . n,D It is shown in [Haw75, Igu03] that, if C is the middle thirds Cantor set, then the • Hausdorff dimension of C (C + t) is 1 log (2) for Lebesgue almost all t in the closed \ 3 3 interval [0, 1] . This is extended to all deleted digits sets in [KP91]. If C is the middle thirds Cantor set, then C (C + t) is self-similar if and only if the • \ 2yk sequence 1 y is “strong periodic”, where t = 1 k and y 1, 0, 1 for { − | k|} k=1 3 k 2 {− } all k by [LYZ11b]. This is extended to sparse C in [PP14].P In particular, C (C + t) \ is not, in general, a self-similar set. Some of the cited papers only consider rational t and some consider Minkowski dimension in place of Hausdorff dimension. It is known, see e.g., [PP12] for an elementary proof, that the (lower) Minkowski dimensions of C (C + t) equals its Hausdorff dimension. \ Palis [Pal87] conjectured that for dynamically defined Cantor sets typically the corre- sponding set F + either has Lebesgue measure zero or contains an interval. The papers [DS08], [MSS09] investigate this problem for random deleted digits sets and solve it in the affirmative in the deterministic case. In two dimensions, where C are certain Sierpinski carpets, the dimension of C (C + t) \ is investigated in [LTZ18] and the dimension is calculated for certain translation vectors t. In [DT07] the authors consider the intersection of a Sierpinski carpet with its rational translates. They use the methods from and obtain results similar to the ones in [NL02]. Both in [LTZ18] and [NL02] the authors study the case where the base is a real number and the “digits” are vectors parallel to the coordinate axes. In this work, we study the case where the base is a complex number of the form n i − ± for an integer n>1 and the set of digits is a proper subset of the set 0, 1,...,n2 .