TY - JOUR
T1 - On the asymptotic formula in Waring's problem with shifts
AU - Biggs, Kirsti D.
PY - 2018/8/1
Y1 - 2018/8/1
N2 - We show that for integers k≥4 and s≥k2+(3k−1)/4, we have an asymptotic formula for the number of solutions to the inequality |(x1−θ1)k+…+(xs−θs)k−τ|<η in positive integers xi, where θi∈(0,1) with θ1 irrational, η∈(0,1], and τ>0 is sufficiently large. We use Freeman's variant of the Davenport–Heilbronn method, along with a new estimate on the Hardy–Littlewood minor arcs, to obtain this improvement on the original result of Chow.
AB - We show that for integers k≥4 and s≥k2+(3k−1)/4, we have an asymptotic formula for the number of solutions to the inequality |(x1−θ1)k+…+(xs−θs)k−τ|<η in positive integers xi, where θi∈(0,1) with θ1 irrational, η∈(0,1], and τ>0 is sufficiently large. We use Freeman's variant of the Davenport–Heilbronn method, along with a new estimate on the Hardy–Littlewood minor arcs, to obtain this improvement on the original result of Chow.
UR - http://www.scopus.com/inward/record.url?scp=85041643711&partnerID=8YFLogxK
U2 - 10.1016/j.jnt.2017.12.009
DO - 10.1016/j.jnt.2017.12.009
M3 - Article (Academic Journal)
AN - SCOPUS:85041643711
SN - 0022-314X
VL - 189
SP - 353
EP - 379
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -