On the Distribution of Solutions to Diophantine Equations

Abstract

Let P be a positive homogeneous polynomial of degree d, with integer coefficients, and for natural numbers $\lambda$ consider the solution sets

$$\displaystyle{Z_{P,\lambda } =\{ m \in \mathbf{Z}^{n}:\ P(m) =\lambda \}.}$$

We’ll study the asymptotic distribution of the images of these sets when projected onto the unit level surface {P = 1} via the dilations, and also when mapped to the flat torus T ⁿ. Assuming the number of variables n is large enough with respect to the degree d we will obtain quantitative estimates on the rate of equi-distribution in terms of upper bounds on the associated discrepancy. Our main tool will be the Hardy-Littlewood method of exponential sums, which will be utilized to obtain asymptotic expansions of the Fourier transform of the solution sets

$$\displaystyle{\omega _{P,\lambda }(\xi ) =\sum _{m\in \mathbf{Z}^{n},\,P(m)=\lambda }e^{2\pi im\cdot \xi }\;,}$$

relating these exponential sums to Fourier transforms of surface carried measures. This will allow us to compare the discrete and continuous case and will be crucial in our estimates on the discrepancy.