On smooth Weyl sums over biquadrates and Waring’s problem

On Certain Character Sums over Smooth Numbers

Weyl Sums over Integers with Linear Digit Restrictions

Abstract. For any given integer q ≥ 2, we consider sets N of non-negative integers that are defined by linear relations between their q-adic digits (for example, the set of non-negative integers such that the number of 1’s equals twice the number of 0’s in the binary representation). The main goal is to prove that the sequence (αn)n∈N is uniformly distributed modulo 1 for all irrational numbers...

Perturbations of Weyl Sums

Write fk(α;X) = ∑ x6X e(α1x + . . . + αkx ) (k > 3). We show that there is a set B ⊆ [0, 1)k−2 of full measure with the property that whenever (α2, . . . , αk−1) ∈ B and X is sufficiently large, then sup (α1,αk)∈[0,1) |fk(α;X)| 6 X1/2+4/(2k−1). For k > 5, this improves on work of Flaminio and Forni, in which a Diophantine condition is imposed on αk, and the exponent of X is 1−2/(3k(k−1)).

Weyl Sums and Atomic Energy Oscillations Weyl Sums and Atomic Energy Oscillations Page 2

We extend Van der Corput's method for exponential sums to study an oscillating term appearing in the quantum theory of large atoms. We obtain an interpretation in terms of classical dynamics and we produce sharp asymptotic upper and lower bounds for the oscillations.

Weyl Sums for Quadratic Roots

The most powerful methods for handling these sums exploit the modern theory of automorphic forms; see [DFI1] for spectral aspects and [DIT] for more arithmetical connections. The sum (1.1) has only a few terms, bounded by the divisor function, so there is not much room for cancellation, but for applications there is a lot of interest in bounds for sums of these as the modulus c varies, say (1.2...