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The proof uses an analogue of residue systems where the residues can take arbitrary real values, and it also requires information about the distribution of primes lying in Bohr sets. We extend a theorem of Rogers to the more general real residues setting, and we extend Siegel–Walfisz’s prime number theorem for arithmetic progressions to the setting of prime numbers lying in Bohr sets. We also prove that circle rotations equidistribute when sampled along such primes, provided the rotation angle is rationally independent of the Bohr set parameter, generalizing a theorem of Vinogradov.</p>"},{"label":"Digital Identifier","value":"m2026-hesseling"},{"label":"Digital Object Identifier (DOI)","value":"https://doi.org/10.14418/twq0s2vd"},{"label":"Digital Origin","value":["born digital"]},{"label":"Discipline","value":["Mathematics"]},{"label":"Date","value":"May 7, 2026"},{"label":"EDTF Date","value":"D:00 M:00 Y:2026"},{"label":"File Format","value":["application/pdf"]},{"label":"Genre","value":["masters theses"]},{"label":"Group","value":["Theses & Dissertations"]},{"label":"Held By","value":"Wesleyan University Library, Wesleyan University, Middletown, CT, USA"},{"label":"Language","value":["English"]},{"label":"Repository","value":["University Archives"]},{"label":"Rights","value":["In Copyright"]},{"label":"Thesis Advisor(s)","value":["Ramírez, Felipe A."]},{"label":"Type","value":["Text"]}],"description":"Prescribing behavior in Duffin–Schaeffer‐type 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