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On a problem of Tamas Varga. (English) Zbl 0787.11002
The authors investigate properties of the expansion of 1 to base \(q\) where \(1 < q < 2\). In part 1 they investigate the class of \(q\) for which the length of consecutive 0 digits in the expansion is not bounded. In part 2 they investigate properties of the sets \(A_n = A_n(q) = \left\{\sum_{i=0}^{n-1}\varepsilon_i q^i,\quad \varepsilon_i = 0\text{ or } 1\right\}\), \(n = 1,2,\dots\) In part 3 they study the digit distribution of the greedy expansion of almost all \(x\), with \(0 < x < 1\) to base \(q\).

MSC:
11A63 Radix representation; digital problems
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
00A07 Problem books
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[1] ERDÖS (P.) , RÉVÉSZ (P.) . - On a problem of Tamás Varga (in Hungarian, Mat. Lapok, t. 24, 1973 , p. 273-282. Zbl 0373.60035 · Zbl 0373.60035
[2] ERDÖS (P.) , RÉVÉSZ (P.) . - On the length of the longest head-run , Topics in information theory, Keszthely (Hungary), Colloq. Math. Soc. János Bolyai, t. 16, 1975 , p. 219-228. MR 57 #17788 | Zbl 0362.60044 · Zbl 0362.60044
[3] ERDÖS (P.) , RÉNYI (A.) . - On a new law of large numbers , J. Analyse Math., t. 22, 1970 , p. 103-111. MR 42 #6907 | Zbl 0225.60015 · Zbl 0225.60015 · doi:10.1007/BF02795493
[4] JOÓ (I.) . - On Riesz bases , Ann. Univ. Sci. Budapest Sect. Math., t. 31, 1988 , p. 141-153. MR 90i:42045 | Zbl 0674.46006 · Zbl 0674.46006
[5] ERDÖS (P.) , HORVÁTH (M.) and JOÓ (I.) . - On the uniqueness of the expansion 1 = \sum q-ni , Acta Math. Hungar., t. 58, 3-4, 1991 . Zbl 0747.11005 · Zbl 0747.11005 · doi:10.1007/BF01903963
[6] ERDÖS (P.) , JOÓ (I.) and KOMORNIK (V.) . - Characterization of the unique expansion 1 = \sum q-ni and related problems , Bull. Soc. Math. France, t. 118, 1990 , p. 377-390. Numdam | Zbl 0721.11005 · Zbl 0721.11005 · numdam:BSMF_1990__118_3_377_0 · eudml:87609
[7] JOÓ (I.) and JOÓ (M.) . - On an arithmetical property of \surd 2 , Publ. Math., t. 39/1-2, 1991 , p. 87-89. MR 92j:11067 | Zbl 0766.11006 · Zbl 0766.11006
[8] ERDÖS (P.) and JOÓ (I.) . - On the expansion 1 = \sum qni , Period. Math. Hungar., t. 23, 1, 1991 , p. 27-30. Zbl 0747.11006 · Zbl 0747.11006 · doi:10.1007/BF02260391
[9] FROUGNY (Ch.) . - Representation of numbers and finite automata . - Math. Systems Theory, to appear.
[10] FROUGNY (Ch.) . - On the expansion of integers in non-integer basis . - Manuscript, 1991 .
[11] RAUZY (G.) . - Sur le développement en base non entière . - Colloque de Théorie des Nombres, Alger, 1989 .
[12] BOGMÉR (A.) , HORVÁTH (M.) and SÖVEGJÁRTÓ (A.) . - On some problems of I. Joó , Acta Math. Hung., t. 58, 1-2, 1991 . MR 92m:11008 | Zbl 0766.11004 · Zbl 0766.11004 · doi:10.1007/BF01903557
[13] RÉNYI (A.) . - Representation for real numbers and their ergodic properties , Acta Math. Acad. Sci. Hungar., t. 8, 1957 , p. 173-179. MR 20 #3843 | Zbl 0079.08901 · Zbl 0079.08901 · doi:10.1007/BF02020331
[14] ERDÖS (P.) and JOÓ (I.) . - On the number of expansions 1 = \sum qni , Preprint of the Math. Inst. of the Hung. Acad. of Sci., n^\circ 30, 1991 .
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