Cataloged from PDF version of article. Thesis (M.S.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2020. Includes bibliographical references (leave 40-42). The properties of compact subsets of the real line which are in the class of Bounded Lebesgue Constants (BLC) are investigated. Knowing that any such set must have 1-dimensional Lebesgue measure zero and nowhere density, and the fact that there are examples of countable sets both inside and outside of the class BLC, families of Cantor-type sets were focused on. Backed up by numerical experiments (up to degree 128) and analytical results, the conjecture “there exists no perfect set in BLC” was put forward. by Yaman Paksoy M.S.