269,052 research outputs found
Restricted divisor sums
There is a body of work in the literature on various restricted sums of the number of divisors of an integer function including that described in [2-9, 11] and summarised in Section 2 below
Continuous functions with universally divergent Fourier series on small subsets of the circle
It is shown that quasi all continuous functions on the unit circle have the
property that, for many small subsets E of the circle, the partial sums of
their Fourier series considered as functions restricted to E exhibit certain
universality properties.Comment: 6 page
Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields
We study the restriction of the Fourier transform to quadratic surfaces in
vector spaces over finite fields. In two dimensions, we obtain the sharp result
by considering the sums of arbitrary two elements in the subset of quadratic
surfaces on two dimensional vector spaces over finite fields. For higher
dimensions, we estimate the decay of the Fourier transform of the
characteristic functions on quadratic surfaces so that we obtain the
Tomas-Stein exponent. Using incidence theorems, we also study the extension
theorems in the restricted settings to sizes of sets in quadratic surfaces.
Estimates for Gauss and Kloosterman sums and their variants play an important
role.Comment: 18 pages, 4 figure
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