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The works of Erd¨os et al. about expansions of 1 with respect to a non-integer base q, referred to as q-expansions, are investigated to determine how far they continue to hold when the number 1 is replaced by a positive number x. It is found that most results about q-expansions for real numbers greater than or equal to 1 are in somewhat opposite direction to those for real numbers less than or equal to 1. The situation when a real number has a unique q-expansion, and when it has exactly two q-expansions are studied. The smallest base number q yielding a unique q-expansion is determined and a particular sequence is shown, in certain sense, to be the smallest sequence whose corresponding base number q yields exactly two q-expansions.
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참고문헌 (7)
참고문헌 신청
K. Dajani / 2002 / From greedy to lazy expansions and their driving dynamics / Expo. Math. 20 (4) : 315 ~ 327
K. Dajani / 2005 / Measures of maximal entropy for random ?-expansions / J. Eur. Math. Soc. 7 (1) : 51 ~ 68
N. Sidorov / 2009 / Expansions in non-integer bases: lower, middle and top orders / J. Number Theory 129 (4) : 741 ~ 754
P. Erd¨os / 1990 / Characterization of the unique expansions 1 =[수식] and related problems / Bull. Soc. Math. France 118 (3) : 377 ~ 390
P. Erd¨os / 1991 / On the uniqueness of the expansions 1 =Σq?ni / Acta Math. Hungar. 58 (3) : 333 ~ 342
K. Dajani / 2005 / Measures of maximal entropy for random ?-expansions / J. Eur. Math. Soc. 7 (1) : 51 ~ 68
N. Sidorov / 2009 / Expansions in non-integer bases: lower, middle and top orders / J. Number Theory 129 (4) : 741 ~ 754
P. Erd¨os / 1990 / Characterization of the unique expansions 1 =[수식] and related problems / Bull. Soc. Math. France 118 (3) : 377 ~ 390
P. Erd¨os / 1991 / On the uniqueness of the expansions 1 =Σq?ni / Acta Math. Hungar. 58 (3) : 333 ~ 342
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