Abstract
For each real irrational number x there exists a unique closest rational integer, which is usually denotedby ((x))o The regularity of the distribution of these closest integers has not yet been studied sufficiently. There are no formulas from which, for an arbitrary x one could finn ((x)). The present paper is devoted to one special case of this problem, which applies to some problems of the theory of Diophantine approximations [I]~ Questions of finding ((ix)), when the decomposition of x as a continued fraction is given and the parameter t runs through a sequence of integers, are considered here. For greater clarity of the calculations it is convenient to have a notation for the numerators and denominators of the convergents, which would explicitly indicate which incomplete parts (or elements) of the continued fraction the corresponding convergent depends on, To this end we use an integral function connected with the Euler-Minding formula. It is based on Euler's algorithm which generalizes the Euclidean algorithm [2]. This function is called the Euler brackets in A. K. Sushkevich [3]. We preserve this name here although it does not appear to be very prevalent.
Published Version
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