http://repository.iiitd.edu.in/xmlui/handle/123456789/1800 Year-2026 Wed, 10 Jun 2026 02:16:47 GMT 2026-06-10T02:16:47Z http://repository.iiitd.edu.in/xmlui/handle/123456789/1982 Topics in the distribution of farey sequences Bittu; Chaubey, Sneha (Advisor) In this thesis, we study the distribution of Farey sequences. Let Q be a positive integer. The Farey sequence FQ of order Q is the set of irreducible fractions in [0, 1] whose denominators do not exceed Q. The first study characterizing the behavior of sequences via equidistribution was carried out in the seminal paper of Weyl. Equidistribution refers to being evenly spaced in a measure space. Equidis- tributed sequences are particularly useful for performing numerical integration. The notion of equidistribution does not provide information about finer statistics, such as randomness, local clustering, and periodic structure of sequences. To study the fine-scale statistics of a sequence, one can study the nearest neighbor gap distri- bution, as well as ν-level correlation measure. We study the equidistribution and correlation measure for Farey sequences. The study of the Farey sequence is of independent interest because of its role in the Diophantine approximation, the circle method, and its connection to the Rie- mann Hypothesis as established by the classical work of Franel and Landau. The Farey fractions of order Q have a one-to-one correspondence with visible lattice points in the triangle with vertices (0, 0), (0, Q), and (Q, Q) through straight lines passing through the origin. The visible lattice points along polynomials have been introduced and studied by Chaubey et al. Motivated by this, we introduce polyno- mial Farey fractions as a subset of fractions a/q ∈ [0, 1] such that the point (a, q) is visible through polynomial curves and examine their distribution. In particular, we study and prove that the lim sup of the pair correlation measure of the polynomial Farey sequence is bounded. For the specific polynomial P (x) = x(x + 1), we show. that the pair correlation measure exists and establish an explicit formula for the pair correlation function which is non-Poissonian. Further, when restricting to prime denominators, the pair correlation measure is shown to be Poissonian. A sequence is said to be Poissonian if it behaves like a random uniformly distributed sequence. It is interesting to study the distribution of Farey fractions with denominators in arithmetic progression, as it is closely related to the Generalized Riemann Hypothesis. Moreover, we study an analog of Chebyshev’s bias question for polynomial Farey fractions with denominators in an arithmetic progression. Chebyshev’s bias question deals with the prime number races and states that there are more primes of the form 4n + 3 than the primes of the form 4n + 1. Furthermore, we study the distribution of the sequence of Farey fractions with k-free denominators lying in an arithmetic progression, denoted by F (m) Q,k . We prove that the sequence F (m) Q,k Q is equidistributed by establishing an estimate for a Weyl sum. Additionally, we establish an equivalent criterion for the Generalized Riemann Hypothesis in terms of the distribution of fractions in F (m) Q,k analogous to the classical results of Franel and Landau. We also investigate the correlation measure of the sequence F (m) Q,k Q and provide an explicit form for the pair correlation measure. Another effective approach to understanding the distribution of the Farey fractions is to examine their indices. We study the distribution of Farey indices by deriving asymptotic formulas for the moments of the index function of Farey fractions with B-free denominators which lie in a given arithmetic progression. Wed, 01 Apr 2026 00:00:00 GMT http://repository.iiitd.edu.in/xmlui/handle/123456789/1982 2026-04-01T00:00:00Z http://repository.iiitd.edu.in/xmlui/handle/123456789/1817 On support and recognition problems for sparse hypergraphs Singh, Karamjeet; Raman, Rajiv (Advisor) A hypergraph H is a pair (V, E), where V is a set of vertices, and E is a collection of subsets of V , called hyperedges. They are used to express complex relations, and they generalize graphs where each element of E is a 2-element subset of V . Hypergraphs are one of the most important combinatorial objects of study in theoretical computer science, and have applications in several domains, including network design, scheduling problems, biology, machine learning, etc. Thus, it is important to study their structural properties. Starting with the work of Zykov [Zyk74], Voloshina and Feinberg [VF84], and John- son and Pollack [JP87], researchers have made several attempts to study the structure of a hypergraph by associating with it an appropriate graph. While their initial attempts were to introduce the planarity of a hypergraph, the notion developed in [VF84; JP87] can be generalized and is now called a support. A support for a hypergraph H = (V, E) is a graph Q = (V, F ) such that for each hyperedge E ∈ E, the induced subgraph Q[E] on the elements of E is connected. With this notion, a hypergraph is considered planar if it admits a support that is a planar graph. The concept of support has practical applications in hypergraph visualization, net- work design, and several optimization problems. Although deciding whether a hyper- graph admits a planar support is NP-hard, identifying sufficient conditions for the existence of such supports, particularly sparse or structured ones, remains a compelling research direction. Most of this thesis delves into the construction of supports for various graph classes. This thesis is divided into three parts. In Part (A), we consider hypergraphs defined by subgraphs of a given host graph. Let G = (V, E) be a graph and H be a collection of subgraphs of G. Then the pair (G, H) naturally defines a hypergraph with vertex set V and a hyperedge V (H) for each H ∈ H. We study support construction in three different settings, depending on whether the host graph G belongs to the class of graphs of (i) bounded genus, (ii) outerplanar, or (iii) bounded treewidth. We gave sufficient conditions that ensure the existence of a support from the same family of graphs as G. The results are extended to dual hypergraphs and to a more general setting- the intersection hypergraphs. We also present a fast algorithm for the construction of a planar support with straight-line embedding when the underlying hypergraph is defined by axis-parallel rectangles and points in R2. Part (B) of the thesis explores the role of supports in solving classical problems such as packing, covering, and coloring problems in hypergraphs. We study these problems for hypergraphs arising from subgraphs of a host graph as well as from geometric regions on orientable surfaces, and present approximation results to the packing and covering problems above. Finally, Part (C) turns to abstract hypergraphs and examines the computational complexity of identifying vertex orderings that forbid fixed patterns. We show NP- hardness of this problem for several vertex orderings, and we deduce implications for the recognition of hypergraphs defined by geometric regions in R2. Tue, 13 Jan 2026 00:00:00 GMT http://repository.iiitd.edu.in/xmlui/handle/123456789/1817 2026-01-13T00:00:00Z