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X-WR-CALNAME:[M2F] Seminar Subin Pulari
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DESCRIPTION:On the Compressibility of Real Numbers: Certain insights using 
 Fourier analytic methods\nSubin Pulari (post-doc M2F)\n\nAbstract: Measuri
 ng the informational content of real numbers has been a significant area o
 f inquiry in algorithmic information theory. Finite-state compressibility 
 (or finite-state dimension) of a real number is a value in [0\, 1] which q
 uantifies the amount of information/randomness in the real number as measu
 red using finite-state automata. Finite-state dimension is the lower asymp
 totic ratio of compression achievable on an infinite string using informat
 ion-lossless finite-state compressors. Interestingly\, the finite-state di
 mension of a real number is also equal to the block Shannon entropy rate o
 f the infinite sequence representing the expansion of the real number in a
  base b. A line of work\, originating from Schnorr and Stimm (1972)\, has 
 established that a number is Borel normal in base b if and only if its bas
 e b expansion has finite-state compressibility equal to 1\, i.e.\, is inco
 mpressible. Hence\, normal numbers are precisely the class of numbers that
  are incompressible using finite-state compressors. \n\nMost of the prior 
 research on finite-state dimension has relied on combinatorial methods. In
  this talk we explore how tools from Fourier analysis can be employed to g
 ain new insights into the compressibility of real numbers\, including the 
 resolution of an open question:\n\n1. One of the most powerful classical t
 ools for investigating normal numbers (numbers having finite-state dimensi
 on 1) is the 1916 Weyl criterion\, which characterizes normality in terms 
 of exponential sums. It was unknown whether Weyl criterion could be genera
 lized from characterizing numbers with finite-state dimension 1 to charact
 erizing numbers with arbitrary finite-state dimensions in [0\,1]. Such a g
 eneralization could make it a quantitative tool for studying data compress
 ion\, prediction\, etc. In this part of the talk\, we generalize the Weyl 
 criterion (1916) for normal numbers to characterize sequences having arbit
 rary finite-state dimension in [0\,1]. We also demonstrate several applica
 tions of this formulation.\n\n2. Absolutely normal numbers\, being finite-
 state incompressible in every base of expansion\, are precisely those numb
 ers which have finite-state dimension equal to 1 in every base. At the oth
 er extreme\, for example\, every rational number has 0 finite-state dimens
 ion in every base. Generalizing this\, Lutz and Mayordomo asked the follow
 ing question: Does there exist any s strictly between 0 and 1 and a real n
 umber r such that r has finite state dimension equal to s in every base?. 
 In this part of the talk\, we use several tools involving exponential sums
  and techniques from Schmidt's work in 1960 to construct\, for any given s
  in (0\,1]\, a real number r having finite-state compressibility equal to 
 s in every base. We thereby answer the open question affirmatively.\n\n\n
 \nImport automatique depuis https://framagenda.org/remote.php/dav/public-c
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DTSTART;TZID=Europe/Paris:20241112T140000
DTEND;TZID=Europe/Paris:20241112T150000
LOCATION:178
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SUMMARY:[M2F] Seminar Subin Pulari
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