TY - JOUR
T1 - On small fractional parts of perturbed polynomials
AU - Minelli, Paolo
N1 - Publisher Copyright:
© 2022 World Scientific Publishing Company.
PY - 2022/9/1
Y1 - 2022/9/1
N2 - Questions concerning small fractional parts of polynomials and pseudo-polynomials have a long history in analytic number theory. In this paper, we improve on the earlier work by Madritsch and Tichy. In particular, let f = P + φ where P is a polynomial of degree k and φ is a linear combination of functions of shape xc, c, 1 < c < k. We prove that for any given irrational ζ we have min 2 ≤ p ≤ Xpprime ζf(p)f,X-ρ(k)+, for P belonging to a certain class of polynomials and with ρ(k) > 0 being an explicitly given rational function in k.
AB - Questions concerning small fractional parts of polynomials and pseudo-polynomials have a long history in analytic number theory. In this paper, we improve on the earlier work by Madritsch and Tichy. In particular, let f = P + φ where P is a polynomial of degree k and φ is a linear combination of functions of shape xc, c, 1 < c < k. We prove that for any given irrational ζ we have min 2 ≤ p ≤ Xpprime ζf(p)f,X-ρ(k)+, for P belonging to a certain class of polynomials and with ρ(k) > 0 being an explicitly given rational function in k.
KW - Diophantine approximation
KW - exponential sums
KW - small fractional parts
UR - http://www.scopus.com/inward/record.url?scp=85132347877&partnerID=8YFLogxK
U2 - 10.1142/S1793042122500853
DO - 10.1142/S1793042122500853
M3 - Article
AN - SCOPUS:85132347877
SN - 1793-0421
VL - 18
SP - 1669
EP - 1690
JO - International Journal of Number Theory
JF - International Journal of Number Theory
IS - 8
ER -