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Character sums over unions of intervals

  • Xuancheng Shao EMAIL logo
Published/Copyright: January 10, 2014
Forum Mathematicum
From the journal Forum Mathematicum Volume 27 Issue 5

Abstract

Let q be a cube-free positive integer and χ(modq) be a non-principal Dirichlet character. Our main result is a Burgess-type estimate for ∑n∈Aχ(n), where A⊂[1,q] is the union of s disjoint intervals I1,...,Is. We obtain a nontrivial estimate for the character sum over A whenever |A|s-1/2>q1/4+ϵ and each interval Ij (1≤j≤s) has length |Ij|>qϵ for any ϵ>0. This follows from an improvement of a mean value Burgess-type estimate studied by Heath-Brown [Number Theory and Related Fields, Springer Proc. Math. Statist. 43, New York (2013), 199–213].

Keywords: Character sum; harmonic analysis
MSC: 11L40

The author is very thankful to his advisor, K. Soundararajan, for proposing the smoothing technique used in the proof of Proposition 1.3 which greatly simplifies the argument, as well as his useful suggestions on exposition, and to I. Shparlinski for pointing to the references [Proc. Steklov Inst. Math. 280 (2013), 67–96; Indag. Math. (2013), DOI 10.1016/j.indag. 2013.02.005].

Received: 2013-5-16
Revised: 2013-10-24
Published Online: 2014-1-10
Published in Print: 2015-9-1

© 2015 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Subgroups which admit extensions of homomorphisms
  3. On the Dedekind completion of function rings
  4. Semiperfect and coreflexive coalgebras
  5. Approximations to the Karcher mean on Hadamard spaces via geometric power means
  6. Separable convolution-elliptic operators with parameters
  7. Charged spaces
  8. Autotopies and quasigroup identities: New aspects of non-associative division algebras
  9. Dualizing complexes and homomorphisms vanishing in Koszul homology
  10. Comparison tests for the asymptotic behaviour of higher-order dynamic equations of neutral type
  11. Hardy spaces associated with a pair of commuting operators
  12. Weighted multilinear Hardy operators and commutators
  13. A characterisation of almost simple groups with socle 2E6(2) or M(22)
  14. The Σ1-invariant for Artin groups of circuit rank 1
  15. The average signature of graph links
  16. Peetre's theorem in the locally convex setting
  17. Small covers, infra-solvmanifolds and curvature
  18. Farrell–Jones spheres and inertia groups of complex projective spaces
  19. Character sums over unions of intervals
  20. Cones of certain isolated left orderings and chain domains
  21. On Wolff's L5/2-Kakeya maximal inequality in ℝ3
  22. Fractional type Marcinkiewicz integral operators on function spaces
  23. Lévy–Khintchine type representation of Dirichlet generators and semi-Dirichlet forms
  24. Homogeneous (α,β)-metrics of Douglas type
https://doi.org/10.1515/forum-2013-0080
Keywords for this article
Character sum; harmonic analysis

Articles in the same Issue

  1. Frontmatter
  2. Subgroups which admit extensions of homomorphisms
  3. On the Dedekind completion of function rings
  4. Semiperfect and coreflexive coalgebras
  5. Approximations to the Karcher mean on Hadamard spaces via geometric power means
  6. Separable convolution-elliptic operators with parameters
  7. Charged spaces
  8. Autotopies and quasigroup identities: New aspects of non-associative division algebras
  9. Dualizing complexes and homomorphisms vanishing in Koszul homology
  10. Comparison tests for the asymptotic behaviour of higher-order dynamic equations of neutral type
  11. Hardy spaces associated with a pair of commuting operators
  12. Weighted multilinear Hardy operators and commutators
  13. A characterisation of almost simple groups with socle 2E6(2) or M(22)
  14. The Σ1-invariant for Artin groups of circuit rank 1
  15. The average signature of graph links
  16. Peetre's theorem in the locally convex setting
  17. Small covers, infra-solvmanifolds and curvature
  18. Farrell–Jones spheres and inertia groups of complex projective spaces
  19. Character sums over unions of intervals
  20. Cones of certain isolated left orderings and chain domains
  21. On Wolff's L5/2-Kakeya maximal inequality in ℝ3
  22. Fractional type Marcinkiewicz integral operators on function spaces
  23. Lévy–Khintchine type representation of Dirichlet generators and semi-Dirichlet forms
  24. Homogeneous (α,β)-metrics of Douglas type
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