TY - JOUR
T1 - Short character sums with beatty sequences
AU - Banks, William D.
AU - Shparlinski, Igor E.
PY - 2006/7
Y1 - 2006/7
N2 - We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence {⌊αn + β⌋: n = 1,2,...}, where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for fixed α,β and a small real number ε > 0, if p is sufficiently large and p 1/3+ε ≤ N ≤ p 1/3+ε, then among the first N elements of the Beatty sequence there are N/2 + o(N) quadratic non-residues modulo p. If additional information about the Diophantine properties of α is available, then the error term o(N) admits a sharper estimate. This work improves and extends several previous results due to M. Z. Garaev and S. N. Preobrazhenskiǐ.
AB - We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence {⌊αn + β⌋: n = 1,2,...}, where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for fixed α,β and a small real number ε > 0, if p is sufficiently large and p 1/3+ε ≤ N ≤ p 1/3+ε, then among the first N elements of the Beatty sequence there are N/2 + o(N) quadratic non-residues modulo p. If additional information about the Diophantine properties of α is available, then the error term o(N) admits a sharper estimate. This work improves and extends several previous results due to M. Z. Garaev and S. N. Preobrazhenskiǐ.
UR - http://www.scopus.com/inward/record.url?scp=33748942083&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:33748942083
SN - 1073-2780
VL - 13
SP - 539
EP - 547
JO - Mathematical Research Letters
JF - Mathematical Research Letters
IS - 4
ER -