On the one-sided boundedness of discrepancy-function of the sequence {nα}

Dupain, Y. and T. Sós, Vera (1980) On the one-sided boundedness of discrepancy-function of the sequence {nα}. Acta Arithmetica, 37. pp. 363-374. ISSN 0065-1036 (print), 1730-6264 (online)

Preview

Text aa37133.pdf Download (403kB) | Preview

Abstract

Let α be an irrational number. If I=[0,β) is a subinterval of [0,1), one puts ΔN(I,α)=card(n:{nα}∈I)−nβ, where {nα} denotes the fractional part of nα. By a result of H. Kesten [same journal 12 (1966/67), 193–212; MR0209253 (35 #155)], it is known that |ΔN(I,α)| is unbounded in N, if β≠{nα} for all n. Continuing earlier work by the first author [Acta Math. Acad. Sci. Hungar. 29 (1977), no. 3–4, 289–303; MR0463131 (57 #3092); Bull. Soc. Math. France 106 (1978), no. 2, 153–159; MR0507746 (80a:10069)], a criterion is given for the one-sided boundedness of the sequence ΔN(I,α): Assume the existence of k,n∈N, r∈Q such that β≡{kα}−r{q2n+1α} mod1, 0≤k<q2n+2, 0≤r≤1, ra2ν∈N for all ν>n (here an denotes the partial quotients of α and qn the denominators of the convergents to α); then ΔN(I,α) is bounded from above. If α has bounded partial quotients, then the converse holds.

Actions (login required)

Edit Item