TY - JOUR
T1 - On a family of sparse exponential sums
AU - Garaev, Moubariz Z.
AU - Rudnick, Zeev
AU - Shparlinski, Igor E.
N1 - Publisher Copyright:
© 2024 The Author(s). Mathematische Nachrichten published by Wiley-VCH GmbH.
PY - 2024/11
Y1 - 2024/11
N2 - We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro-geometric methods such as Weil's bounds, Bourgain gave a nontrivial estimate for these and more general sums. In this work, we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.
AB - We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro-geometric methods such as Weil's bounds, Bourgain gave a nontrivial estimate for these and more general sums. In this work, we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.
KW - binomial exponential sums
KW - moments
KW - trinomial exponential sums
UR - https://www.scopus.com/pages/publications/85203304156
U2 - 10.1002/mana.202300426
DO - 10.1002/mana.202300426
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:85203304156
SN - 0025-584X
VL - 297
SP - 4214
EP - 4231
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
IS - 11
ER -