Minimum and maximum order of magnitude of the discrepancy of (nα)

On Sums of Fractional Parts {nα+γ}

“…It is perhaps worth recalling here the theorem of Kesten [10]; namely that if the partial quotients of : are bounded, then 2 m (1&#, :) is bounded if and only if #=[n:] for some integer n (equivalently c i =0 for all but finitely many i). Notice that, varying both m and #, [1,16,17,19]. The right-hand sides in (i) are in this case attained for m $ # := i odd (a i &c i ) q i&1 and m " # := i even c i q i&1 respectively, those in (ii) are again achieved for the #$ m and #" m of (10).…”

“…There is at least one major difference though; it is no longer true that bounded quotients are sufficient to cause the sizeable positive negative swings that occur when #=0. For example (as we shall show in a subsequent paper), if :=-2 one-sidedly bounded sums C m (-2, 1 2 )>0 occur when #=1Â2. However, such a gamma should be thought of as exceptional.…”

“…Using m ( j) to denote the sum of the odd indexed, i odd z i q i&1 , or even indexed, i even z i q i&1 , terms of the Zeckendorff representation of m= z i q i&1 , as j is odd or even respectively, we note that #$ m =[&m (1) :] and #" m =[&m (0) :]. In particular C m (:, #) can in these cases be rewritten as a sum of wholly positive or wholly negative terms; …”

Combinatorics, Words and Symbolic Dynamics

“…For three-distance truncations, there does not exist a formula which expresses ∆ T for a truncation of index k solely with T, q k , q k−1 , θ k , θ k+1 . However, the paper (Baxa and Schoissengeier, 1994) provides estimates that relate the discrepancy ∆ T (α) for an index k to the average A k := (1/k) ∑ k i=1 m i of the first k digits in the continued fraction expansion of α. We are thus led to consider constrained models of another type, which deal with the sets I [M] of real numbers for which each average A k is bounded by some constant M. We may also consider their rational counterparts Ω [M] N .…”

Répartition des suites (nα)n∈Net substitutions

“…(Ici {x} = x− x désigne la partie fractionnaire de x.) De nombreux auteurs ontétudié D * N (α) pour α irrationnel, notamment pour les plus récents citons : C. Baxa [2,3], C. Baxa et J. Schoissengeier [4], Y. Dupain [9], Y. Dupain et V. T. Sós [11], J. Lesca [14], L. Ramshaw [19], H. Niederreiter [15], J. Schoissengeier [23,24] et V. T. Sós [27]. De nombreuses références complémentaires pourrontêtre trouvées dans l'ouvrage de L. Kuipers et H. Niederreiter [13] et plus récemment dans celui de M. Drmota et R. F. Tichy [6].…”