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Bulletin of the London Mathematical Society Advance Access originally published online on March 11, 2008
Bulletin of the London Mathematical Society 2008 40(1):88-96; doi:10.1112/blms/bdm111
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© 2008 London Mathematical Society

Density of non-residues in Burgess-type intervals and applications

W. D. Banks

Department of Mathematics
University of Missouri
Columbia, MO 65211
USA
bbanks@math.missouri.edu

M. Z. Garaev

Instituto de Matemáticas
Universidad Nacional Autónoma de México
C.P. 58089 Morelia
Michoacán
México

D. R. Heath-Brown

Mathematical Institute
24-29, St Giles’
Oxford
OX1 3LB
UK
rhb@maths.ox.ac.uk

I. E. Shparlinski

Department of Computing
Macquarie University
Sydney NSW 2109
Australia
igor@ics.mq.edu.au

Received 18 January 2007. Revision received 3 September 2007.

We show that for any fixed {varepsilon} > 0, there are numbers {delta} > 0 and p0 ≥ 2 with the following property: for every prime p ≥ p0 and every integer N such that p1/(4{surd}e )+{varepsilon} ≤ N ≤ p, the sequence 1, 2, ..., N contains at least {delta} N quadratic non-residues modulo p. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski-Shapiro sequences.

2000 Mathematics Subject Classification 11A15, 11L40, 11N37.


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