Exponential sums and a conjecture involving quantization of bandlimited functions

Abstract: Sigma-delta modulation is a classical method for oversampled coarse quantization which enables approximation of bandlimited functions (e.g. audio signals) at high sampling rates despite using only two fixed levels to round each sample. In the basic form of this method (the "first order" case), the approximation rate is $\lambda^{-1}$ in the uniform norm where $\lambda$ denotes the oversampling ratio, but the pointwise error has been shown to decay at least at the rate $\lambda^{-4/3+\epsilon}$ under generic conditions. Meanwhile, a long-standing folklore conjecture based on numerical simulations predicts square-root cancellation "on average", i.e. approximation rate of order $\lambda^{-3/2+\epsilon}$. We disprove the conjecture for the Besicovitch norm, utilizing certain exponential sums of bandlimited phase. Joint work with Maksym Radziwill.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)