Estimation for smooth Weyl's sums

Let $k \in \N$ and $P$ be large real numbers. When $2 \le R \le P$, define the set of $R$-smooth numbers: \[ \A(P,R) = \{n \in [1,P] \cap \Z : p|n \Rightarrow p \le R\} \] Let $\alpha \in \R$. We define the smooth Weyl sum: \[ f(\alpha; P, R) = \sum_{x \in \A(P,R)} e(\alpha x^k). \] In this note, I will explain only the following theorem, from which the bound for Waring's Problem is established using the classical circle method by Hardy and Littlewood.

Theorem 1

Let $\m$ denote the set of real numbers $\alpha$ s.t. whenever $a \in \Z$, $q \in \N$, $(a,q)=1$ and $|\alpha - \frac{a}{q}| \le q^{-1}P^{1-k}$, one has $q > P$. Then when $\eta = \eta(\epsilon, k)$ is a sufficiently small positive number, and $2 \le R \le P^\eta$, we have \[ \sup_{\alpha \in \m} |f(\alpha; P, R)| \ll_{\epsilon, k} P^{1-\rho(k)+\epsilon}, \] where, when $k$ is large, $\rho(k)^{-1} = k(\log k + O(\log \log k))$.

We define $S_s(P,R)$ to be the number of solutions of the diophantine equation \[ x_1^k + \dots + x_s^k = y_1^k + \dots + y_s^k \] with $x_i, y_i \in \A(P,R)$ ($1 \le i \le s$). Thus, \[ S_s(P,R) := \int_0^1 |f(\alpha; P, R)|^{2s} d\alpha. \]

The estimation for $f(\alpha; P, R)$ is based on Vaughan in Section 10 of "A new iterative method in Waring's Problem". It connects the smooth Weyl's sum to $S_s(P,R)$, and by working on mean values, we obtain the bound for $S_s(P,R)$. Now, I will introduce the above as the main theorem.

We say $\Delta_s = \Delta_{s,k}$ is permissible if $S_s(P,R) \ll P^{\lambda_{s,k}+\epsilon}$ with $\lambda_{s,k} = 2s - k + \Delta_{s,k}$. Then, if it is permissible, we have $\Delta_s$ non-negative and $\Delta_s \le k$.

Theorem 2

Let $k \ge 4$ and $t \in \N$. For each $s \in \N$ with $2 \le s \le t$, define $\Delta_s = \Delta_{s,k}$ to be the unique positive solution of the equation \[ \Delta_s e^{\Delta_s/k} = k e^{1-2s/k}. \] Then, $\Delta_{s,k}$ is permissible and then $\Delta_{s,k}^* = k e^{1-2s/k}$ is permissible.

Theorem 3

Let $\alpha \in \R$ and $r \in \mathbb{N}$, and suppose that $\mathrm{Q}, M$ and $R$ satisfy $2 \le R \le M < \mathrm{Q}$. Then \[ \sum_{\substack{x \in \A(\mathrm{Q},R) \\ (x,r)=1}} \dots \ll R \log \mathrm{Q} \sup \dots \] (Note: The explicit bound involves auxiliary sums defined below)

where \[ V_r(\alpha; Q, M, R; \pi, \theta) = \sum_{\substack{v \in \mathcal{B}(M, \pi, R) \\ (v,r)=1}} \left| \sum_{\substack{u \in \A(Q/M, \pi) \\ (u,r)=1}} e(\alpha (uv)^k + \Theta u) \right|, \] \[ \mathcal{B}(M, \pi, R) = \{v \in \N : M < v \le M\pi, \pi|v, \text{ and } p|v \Rightarrow \pi \le p \le R\}. \]

For an integer $v$, let $S_0(v)$ denote the largest square-free divisor of $v$. Define the set $C_q(l)$ by \[ C_q(l) = \{x \in \Z \cap [1,l] : S_0(x) | S_0(q)\}. \]

Lemma 1

Suppose that $L$ is a positive real number and $r$ is a positive integer with $\log r \ll \log L$. Then for each $\epsilon > 0$, \[ \text{Card } C_r(L) \ll L^\epsilon. \]

Proof:

Write $l = \#\{y \le L : S_0(y) = S_0(r)\}$, then \[ \# C_r(L) \ll d(L) \#\{y \le L : S_0(y) = S_0(r)\} \ll L^\epsilon l. \] Suppose that $S_0(r) = p_1 \dots p_n$, then $n \ll \frac{\log r}{\log \log r} \ll \frac{\log L}{\log \log L}$. Note $l$ is bounded above by # of solutions of the inequality \[ u_1 \log p_1 + \dots + u_n \log p_n \le \log L \] with $u_i \in \N$. But for $m \in \N \cap [1, \log L]$, # of solutions of the equation $u_1 + \dots + u_n = m$ with $u_i \in \N \cup \{0\}$ is $\ll \frac{(m+n)^{n-1}}{(n-1)!}$. So that $l \ll \frac{(\log L)^n}{(n-1)!}$. By Stirling's formula, it yields that $\log l \ll n(\log \log n - \log n) + n$. Hence, by combining with the previous bound for $n$, we complete the proof of the lemma. ◼

The author then estimates $f(\alpha; P, R)$ in large moduli and small moduli, respectively.

Lemma 2

Suppose that $\lambda$ satisfies $\frac{1}{2} < \lambda < 1$, and write $M = P^\lambda$. Let $\alpha \in \R$, and suppose that there exists $a \in \Z$ and $q \in \N$ satisfying $(a,q)=1$ and $|\alpha - \frac{a}{q}| \le q^{-2}$. Then when $t, w \in \N$ and $\Delta_t$ and $\Delta_w$ are permissible: \[ f(\alpha; P, R) \ll q^\epsilon P^{1+\epsilon}(M^{\Delta_w}(P/M)^{\Delta_t}(q^{-1} + M^{-k} + (P/M)^{-k} + qP^{-k}))^{\frac{1}{2tw}} + M. \]

Proof:

By applying Theorem 3 with $r=1$, $\exists \pi$ prime to $\pi \le R$ and $\theta \in [0,1)$, such that \[ f(\alpha; P, R) \ll q^\epsilon P^\epsilon R \sum_{v \in \A(MR,R)} |h(\alpha; v, \theta)| + M \] where $$h(\alpha; v, \theta) = \sum_{v \in \A(P/M, \pi)} e(\alpha u^k v^k + \theta u)$$ After taking this sum to $t$-th power, it counts the solutions of the diophantine equation $$u_1^k + \dots + u_t^k = c$$ with $u_i \in \A(P/M, \pi)$ weighted as $e(\theta(u_1+\dots+u_t))$, where $$u_1+\dots+u_t = c \in [1, t(P/M)^k].$$ Let $n_c$ be the number of solutions of the above without counting with weight, then \[ (\sum_v |h(\dots)|)^{2tw} \le (MR)^{2w(t-1)} (\sum_c n_c)^{2w-2} (\sum_c n_c^2) J_w(\alpha) \] where $$J_w(\alpha) = \sum_c |g(\alpha; c, \theta)|^{2w}$$ and $$g(\alpha; c, \theta) = \sum_v \epsilon(v, \theta) e(\alpha c v^k).$$ By the underlying diophantine equations of $n_c$, we get $\sum n_c \le (P/M)^t$ and $\sum n_c^2 \le S_t(P/M, R)$. If we take $$\tilde{n}_d = \int_0^1 |g(\beta, c, \theta)|^{2w} e(-\beta d) d\beta,$$ clearly $\tilde{n}_d \le S_w(MR, R)$. Therefore, we can write \[ J_w(\alpha) = \sum_{c} \sum_{d} \tilde{n}_d e(\alpha c d) \] whence we deduce its bound by applying Weyl's classical inequality. Then, after applying the bound established in Theorem 2, we complete the proof. ◼

As for the small moduli, apart from adapting the idea of connecting the power of Weyl's sum with solutions to diophantine equations, we utilize the benefits of small moduli. This yields a well-spaced condition, allowing us to apply the large Sieve inequality to split the cardinality of solution sets and the exponential sums. (This is a method by Vaughan in the same paper where he established Theorem 3).

Lemma 3

Suppose that $\frac{1}{2} < \lambda < 1$ and write $M = P^\lambda$. Let $\alpha \in \R$, $a \in \Z$ and $q \in \N$ satisfy $(a,q)=1$, $|q\alpha - a| \le \frac{1}{2}(MR)^{-k}$. Suppose that $q \le 2(MR)^k$ and either $|q\alpha - a| > M P^{-k}$ or $q > MR$. Then if $s \in \N$ with $2s \ge k+1$, and $\Delta_s$ is permissible, \[ f(\alpha; P, R) \ll M^{1+\epsilon} + P^{1+\epsilon} (M^{-1}(P/M)^{\Delta_s}(1 + q(P/M)^{-k}))^{\frac{1}{2s}}. \]

Proof:

Observe that \[ f(\alpha; P, R) = \sum_{d \in C_q(P) \cap \A(P,R)} \sum_{\substack{x \in \A(P/d, R) \\ (x,q)=1}} e(\alpha (xd)^k). \] On applying Lemma 1, we can apply Theorem 3 for some $\theta \in [0,1)$ and a prime $\Pi \le R$, which yields \[ f(\alpha; P, R) \ll M^{1+\epsilon} + P^\epsilon R \cdot g(\alpha; d, \pi, \theta), \] where $g(\dots)$ is the inner sum. Therefore, by raising the LHS side to $2s$ power and let $b_y$ denote # of solutions to the diophantine equation $$u_1^k + \dots + u_s^k = y$$ with $u_i \in \A(P/M, R)$, it follows that \[ g(\dots)^{2s} \ll \left(\frac{MR}{d}\right)^{2s-1} \sum_{v} \left| \sum_{y} b_y e(\alpha v^k y) \right|^2. \] We then classify $v$ into $O(P^\epsilon d^k)$ sets $V_j$, so that for $v_1, v_2 \in V_j$, we have $(v_1 d)^k \equiv (v_2 d)^k \pmod q$. Thus, there is a suitable $j$ such that we manage to show $(P/M)^{-k}$ is well-spaced. Then, by the large sieve inequality, \[ \sum_{v \in V_j} \left| \sum_{y} b_y e(\alpha (vd)^k y) \right|^2 \ll (q + (P/M)^k) \sum |b_y|^2 \ll (q+(P/M)^k) S_s(P/M, R). \] Then the lemma follows from the bound in Theorem 2. ◼

Therefore, by combining Lemma 2 and Lemma 3 and choosing some suitable parameters, we obtain Theorem 1. Hence, it leaves us to verify that Theorem 2 and 3 are valid. Theorem 2 is a simplification of the following by Trevor Wooley.

Theorem 4

Suppose that $t$ is a positive integer and $\mu$ is a positive real number with $2t-k < \mu \le 2t$ and satisfying the property that, given $\epsilon > 0$, there is a positive number $\eta_0 = \eta_0(k,s)$ s.t. whenever $0 < \eta < \eta_0$, then $S_t(P, P^\eta) \ll P^{\mu + \epsilon}$. Define the real numbers $\lambda_s, \theta_s, \Delta(s)$ ($s=t, t+1, \dots$) successively by $\lambda_t = \mu, \theta_t = 0, \Delta(t) = \mu - 2t + k$, and for $s > t$, by \[ \theta_s = \frac{1}{k+\Delta(s-1)} + \left(\frac{1}{k} - \frac{1}{k+\Delta(s-1)}\right) \left(\frac{k-\Delta(s-1)}{2k}\right)^{k-1} \] \[ \Delta(s) = \Delta(s-1)(1-\theta_s) + k\theta_s - 1 \] \[ \lambda_s = 2s - k + \Delta(s). \] Then given $t'$ and $\epsilon' > 0$, there is an $\eta_1 = \eta_1(k, \epsilon', t')$ such that whenever $0 < \eta < \eta_1$ and $t \le s \le t'$, \[ S_s(P, P^\eta) \ll P^{\lambda_s + \epsilon'}. \]

Let $\Psi(z,c)$ denote a polynomial with integer coefficients in the variables $z, c_1, \dots, c_t$ of degree in least one in $z$. Let $S_s(P,Q,R) = S_s(P,Q,R; \Psi, C, C')$ denote the number of solutions of the equation \[ \Psi(z, c) + x_1^k + \dots + x_s^k = \Psi(z', c') + y_1^k + \dots + y_s^k. \] For a given real number $1 \le P^\theta \le Q$, let $T_s(P, Q, R; \theta)$ denote the number of solutions to the equation \[ \Psi(z, c) + w^k(u_1^k + \dots + u_s^k) = \Psi(z', c) + w^k(v_1^k + \dots + v_s^k). \] The idea of the proof of Theorem 4 is as follows:

Bound $S_s(P,Q,R)$ with terms relating to $T_s$.
Replace $\Psi(z,c)$ with a recursive function.
Relate $T_s$ to $S_s$.

Proof of Theorem 3 (by Vaughan): Write \[ f(\alpha; P, R) = \sum_{q_0|q} \sum_{\substack{x \in \A(P,R) \\ (x^k, q) = q_0}} e(\alpha x^k). \] Let $q_k^k$ be the largest $k$-th power dividing $q$. Then $q_0 = q_k^k \dots q_1$. Hence, \[ f(\alpha; P, R) = \sum_{q,r}^* T(\dots) + O(q^\epsilon P^{1-\delta}) \] where $\sum^*$ indicates the sum over $q_0 r = q$ and square-free conditions.

Lemma

Suppose that $2 \le R \le M < y \le C$ and $y \in \A(Q,R)$. Then there exists a unique triple $(p, u, v)$ with (i) $y = uv$, (ii) $u \in \A(Q/v, p)$, (iii) $M < v \le Mp$, (iv) $p|v$, (v) $p'|v \Rightarrow p \le p' \le R$.

And for any pair of $(p, u, v)$ satisfying (i)-(v), we have $uv \in \A(Q, R)$ and $M < uv \le C$. Note $p$ depends on $v$. That is, we can form a bijection between $y$ and $(p, u, v)$. Hence, deduce Theorem 3 by replacing $y$ with $uv$.

This completes the proof of the main Theorem.