BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Colin Faverjon (CNRS\, Université de Picardie Jules Verne) DTSTART:20251104T130000Z DTEND:20251104T140000Z DTSTAMP:20260423T021234Z UID:OWNS/155 DESCRIPTION:Title: W riting numbers in multiple bases: the viewpoint of finite automata\nby Colin Faverjon (CNRS\, Université de Picardie Jules Verne) as part of On e World Numeration seminar\n\n\nAbstract\nAlthough binary and decimal repr esentations of numbers coexist seamlessly in our digital world\, these cha nges of basis conceal profound mysteries. Consider\, for example\, the fol lowing statements\, both currently out of reach:
\n- The date when Tru mp will leave the US presidency appears in the decimal expansion of every sufficiently large power of 2\;
\n- The real number whose binary expan sion is the characteristic sequence of powers of 3 contains the pattern 13 12 infinitely often in its decimal expansion.\n
\nBoth statements rely on the heuristic that expansions in multiplicatively independent bases (su ch as 2 and 10) should share no common structure. Furstenberg captured thi s heuristic through a series of results and conjectures concerning the joi nt behavior of the dynamical systems ×p and ×q on the torus.\n\nIn this talk\, we approach this question from the perspective pioneered by Turing\ , Hartmanis\, Stearns\, and Cobham: that of computational complexity. Powe rs of 2 are particularly easy to recognize from their base-2 expansion—a task achievable by a finite automaton. Cobham's theorem then implies that no automaton can recognize them from their decimal expansion. Similarly\, one can readily construct a finite automaton with output that produces th e binary expansion of the real number introduced above. Whether there exis ts another automaton producing its decimal expansion remained open until r ecently.\n\nIn this talk\, we present how this question has been solved us ing a transcendence method known as Mahler's method. While this approach y ields a new proof and an algebraic generalization of Cobham's theorem\, it s main contribution is the following statement: no irrational real numb er has expansions in two multiplicatively independent bases that can both be produced by finite automata.\n LOCATION:https://researchseminars.org/talk/OWNS/155/ END:VEVENT END:VCALENDAR