The sum of digits of \lfloor n^c\rfloor

Subsequences of automatic sequences indexed by⌊nc⌋and correlations

“…For specific values of c, Mauduit and Rivat proved in [11,12] the following result for the sum-of-digits function s q in base q: For m 2 and for any c in (1, 7/5), the density with which a residue a modulo m is observed in the sequence (s q (⌊n c ⌋)) n∈N exists and is equal to 1/m. (Compare with [16], where the same result is shown for all c ∈ R \ N provided that the base q is large enough (depending on c). )…”

Piatetski-Shapiro sequences via Beatty sequences

“…Note that by the same orthogonality argument (4) can be deduced from from (18), which gives an alternative to Kim's proof [11].…”

“…A more basic question is to ask for the distribution of ⌊n c ⌋ in residue classes. In this case it is known that for all noninteger c > 1, all positive integers m and all a ∈ Z we have |{n ≤ x : ⌊n c ⌋ ≡ a mod m}| = x m + O x 1−ε for some ε = ε(c) that can be given explicitly, see Deshouillers [6] and Morgenbesser [18]. Another line of research was initiated by Mauduit and Rivat [13] which concerns the behaviour of q-multiplicative functions on Piatetski-Shapiro sequences.…”

Primes in floor function sets

“…There is an extensive body of research on arithmetic functions with integer parts of real-valued functions, most commonly, with Beatty ⌊αn + β⌋ sequences, see, for example, [1,3,6,12,13], and Piatetski-Shapiro ⌊n γ ⌋ sequences, see, for example, [2,4,5,7,17,20], with real α, β and γ.…”