Waring's Problem and the Circle Method

Waring's problem asks: for each $k \geq 2$ , what is the smallest $s$ such that every sufficiently large integer is a sum of $s$ perfect $k$ -th powers? The Hardy-Littlewood circle method provides the primary tool for this and related additive problems.

Waring's Problem

Definition8.1Waring's Problem

Let $g(k)$ be the smallest $s$ such that every positive integer is a sum of $s$ $k$ -th powers, and $G(k)$ the smallest $s$ such that every sufficiently large integer is a sum of $s$ $k$ -th powers. Classical results: $g(2) = 4$ (Lagrange), $g(3) = 9$ (known), $g(4) = 19$ . Asymptotically: $G(k) \leq (1 + o(1))k\log k$ (Vinogradov-Wooley). Known: $G(2) = 4$ , $G(3) \leq 7$ (likely 4), $G(4) \leq 15$ .

Definition8.2The Hardy-Littlewood Circle Method

To count representations $R_s(N) = |\{(x_1,\ldots,x_s) : x_1^k + \cdots + x_s^k = N, x_i \geq 0\}|$ , write $R_s(N) = \int_0^1 f(\alpha)^s e^{-2\pi i N\alpha}\,d\alpha$ where $f(\alpha) = \sum_{1 \leq n \leq N^{1/k}} e^{2\pi i n^k \alpha}$ is the Weyl sum. The circle method partitions $[0,1]$ into major arcs $\mathfrak{M}$ (near rationals $a/q$ with small $q$ ) and minor arcs $\mathfrak{m}$ (the complement).

Major and Minor Arcs

ExampleMajor Arc Contribution

On major arcs near $a/q$ with $(a,q)=1$ , $q \leq P$ : $f(\alpha) \approx \frac{1}{q}S(q,a)\int_0^{N^{1/k}} e^{2\pi i\beta t^k}dt$ where $S(q,a) = \sum_{r=1}^q e^{2\pi i ar^k/q}$ is a complete exponential sum and $\beta = \alpha - a/q$ is small. The integral of $f^s e^{-2\pi iN\alpha}$ over $\mathfrak{M}$ gives the singular series $\mathfrak{S}(N) = \sum_{q=1}^\infty \sum_{(a,q)=1} q^{-s}S(q,a)^s e^{-2\pi i aN/q}$ times the singular integral $\mathfrak{J}(N) = \int_\mathbb{R}(\int_0^{N^{1/k}} e^{2\pi i\beta t^k}dt)^s e^{-2\pi i N\beta}d\beta$ . When $s$ is large enough, $\mathfrak{S}(N) > 0$ and $\mathfrak{J}(N) \asymp N^{s/k-1}$ .

RemarkMinor Arc Bounds

On minor arcs, one needs $|f(\alpha)|$ to be small on average. Weyl's inequality gives $|f(\alpha)| \ll N^{1/k+\varepsilon}|f(\alpha)|^{-1} \cdot (\text{main term bound})$ . For $s$ large enough (depending on $k$ ), the minor arc contribution is $o(N^{s/k-1})$ , and $R_s(N) \sim \mathfrak{S}(N)\mathfrak{J}(N) > 0$ for all large $N$ .