Browsing by Author "Elofsson, Carl"

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    The Banach-Tarski paradox
    (2022-07-04) Elofsson, Carl; Nilsson, Adrian; Söderberg, Nicolas; Westlund, Tim; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaper
    In this thesis we present a proof of the Banach-Tarski paradox, a counterintuitive result that states that any ball in R3 can be cut into finitely many pieces and then be reassembled into two copies of the original ball. Since the result follows from the axiom of choice it is important for assessing its role as an axiom of mathematics. A related result that we also include is that the minimal number of pieces in such a decomposition of any ball in R3 is five. The proof uses the paradoxicality of the free group on two generators and the existence of a free subgroup of the special orthogonal group SO3. We also give a proof of Tarski’s theorem, which states that the existence of a finitely additive, isometry invariant measure normalizing a set is equivalent to that set not being paradoxical. The proof makes use of the Hahn-Banach theorem and relies on the concept of a group acting on several copies of a set.
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    Prime number races
    (2024-08-12) Elofsson, Carl; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaper
    In this thesis we investigate the behaviour of primes in arithmetic progressions, with a focus on the phenomenon known as Chebyshev’s bias. Under the assumption of the Generalized Riemann Hypothesis and the Linear Independence Hypothesis, we prove that there is a bias towards quadratic non-residues. Additionally we extend the investigation to the setting of function fields. In the function field setting, we investigate the behaviour of prime polynomials in residue classes modulo a fixed monic polynomial. Moreover, we prove that for an irreducible polynomial m there is a bias towards quadratic non-residues modulo m.