[proofplan] We apply Weyl differencing $k-1$ times, reducing the degree-$k$ polynomial phase to linear phases whose coefficients are controlled by the leading coefficient $\alpha_k$. The iterated finite difference has leading linear coefficient $k!\alpha_k h_1\cdots h_{k-1}$, so after grouping difference tuples by their product the divisor bound reduces the problem to estimating sums of $\min\{X,\|k!h\alpha_k\|_{\mathbb R/\mathbb Z}^{-1}\}$. The rational approximation $\alpha_k\approx a/q$ gives the standard Vinogradov summation estimate for these quantities. Substituting that estimate into the differenced inequality and taking the $2^{k-1}$-st root gives the claimed bound. [/proofplan]

[step:Apply Weyl differencing until only linear phases remain]Set $N:=\lfloor X\rfloor$ and define \begin{align*} S:=\sum_{1\le n\le N}e(P(n)). \end{align*} For $h\in\mathbb Z$ and a function $Q:\mathbb Z\to\mathbb R$, define the finite difference \begin{align*} \delta_h Q:\mathbb Z\to\mathbb R,\qquad (\delta_hQ)(n):=Q(n+h)-Q(n). \end{align*} Iterating the [[Weyl Differencing Lemma](/theorems/9052)][citetheorem:9052] $k-1$ times, with all intervals produced by the successive restrictions retained explicitly, gives \begin{align*} |S|^{2^{k-1}}\lesssim_k X^{2^{k-1}-k}\sum_{\substack{h_1, \dots, h_{k-1}\in\mathbb Z, |h_i|<X}}\left|\sum_{n\in I(h_1,\dots,h_{k-1})}e((\delta_{h_{k-1}}\cdots\delta_{h_1}P)(n))\right|, \end{align*} where, for each tuple $(h_1,\dots,h_{k-1})$, the set $I(h_1,\dots,h_{k-1})\subset\mathbb Z$ is an integer interval of cardinality at most $N$. Define the product map \begin{align*} H:\mathbb Z^{k-1}\to\mathbb Z,\qquad (h_1,\dots,h_{k-1})\mapsto h_1\cdots h_{k-1} \end{align*}. By the [[Leading Coefficient of an Iterated Finite Difference](/theorems/9053)][citetheorem:9053], whenever $H(h_1,\dots,h_{k-1})\ne 0$, the polynomial \begin{align*} n\mapsto(\delta_{h_{k-1}}\cdots\delta_{h_1}P)(n) \end{align*} has degree $1$ and leading coefficient \begin{align*} k!\alpha_k H(h_1,\dots,h_{k-1}). \end{align*}[/step]

[guided]Set $N:=\lfloor X\rfloor$ and write the Weyl sum as \begin{align*} S:=\sum_{1\le n\le N}e(P(n)). \end{align*} The purpose of Weyl differencing is to trade one power of $|S|$ for a family of sums whose phases are finite differences of $P$. Since each finite difference lowers the degree of a polynomial by one when the difference parameter is nonzero, applying the process $k-1$ times turns a degree-$k$ phase into a linear phase. For $h\in\mathbb Z$ and a function $Q:\mathbb Z\to\mathbb R$, define \begin{align*} \delta_h Q:\mathbb Z\to\mathbb R,\qquad (\delta_hQ)(n):=Q(n+h)-Q(n). \end{align*} We apply the [Weyl Differencing Lemma][citetheorem:9052] successively. At each stage the summation interval is restricted so that both shifted arguments remain in the preceding interval; hence after $k-1$ applications the inner summation is over an integer interval $I(h_1,\dots,h_{k-1})\subset\mathbb Z$ with at most $N$ elements. The iteration gives \begin{align*} |S|^{2^{k-1}}\lesssim_k X^{2^{k-1}-k}\sum_{\substack{h_1, \dots, h_{k-1}\in\mathbb Z, |h_i|<X}}\left|\sum_{n\in I(h_1,\dots,h_{k-1})}e((\delta_{h_{k-1}}\cdots\delta_{h_1}P)(n))\right|. \end{align*} The factor $X^{2^{k-1}-k}$ is the accumulated normalization from the $k-1$ differencing steps. Now define the product map $H:\mathbb Z^{k-1}\to\mathbb Z$ by \begin{align*} H(h_1,\dots,h_{k-1}):=h_1\cdots h_{k-1}. \end{align*} When this product is nonzero, each difference step genuinely lowers the degree by one. The [Leading Coefficient of an Iterated Finite Difference][citetheorem:9053] applies to the polynomial $P$ over the characteristic-zero field $\mathbb R$ and gives that \begin{align*} n\mapsto(\delta_{h_{k-1}}\cdots\delta_{h_1}P)(n) \end{align*} is linear with leading coefficient \begin{align*} k!\alpha_k H(h_1,\dots,h_{k-1}). \end{align*} This is the key structural point: all lower coefficients of $P$ affect only the constant term of the final linear phase, while the oscillation of the final sum is governed by $k!\alpha_k h_1\cdots h_{k-1}$.[/guided]

[step:Bound the resulting linear sums by distance to the nearest integer] Let $D:=k!$. For $t\in\mathbb R$, define the distance-to-integer map \begin{align*} \|\cdot\|_{\mathbb R/\mathbb Z}:\mathbb R\to\left[0,\tfrac12\right],\qquad t\mapsto \operatorname{dist}(t,\mathbb Z)=\inf_{m\in\mathbb Z}|t-m| \end{align*}. If $H(h_1,\dots,h_{k-1})=0$, the corresponding inner sum has absolute value at most $N\le X$. There are $\lesssim_k X^{k-2}$ such tuples, so the total contribution of these tuples is $\lesssim_k X^{k-1}$. Assume now that $H(h_1,\dots,h_{k-1})\ne 0$. There exists $c(h_1,\dots,h_{k-1})\in\mathbb R$ such that, for all $n\in\mathbb Z$, \begin{align*} (\delta_{h_{k-1}}\cdots\delta_{h_1}P)(n)=D\alpha_kH(h_1,\dots,h_{k-1})n+c(h_1,\dots,h_{k-1}). \end{align*} For every $\theta\in\mathbb R$, every $\gamma\in\mathbb R$, and every integer interval $J\subset\mathbb Z$ with $\#J\le X$, the finite geometric-series identity gives \begin{align*} \left|\sum_{n\in J}e(\theta n+\gamma)\right|\le \min\left\{X,\frac{1}{2\|\theta\|_{\mathbb R/\mathbb Z}}\right\}, \end{align*} with the convention that the second term is $+\infty$ when $\theta\in\mathbb Z$. Using this with $\theta=D\alpha_kH(h_1,\dots,h_{k-1})$, we obtain \begin{align*} |S|^{2^{k-1}}\lesssim_k X^{2^{k-1}-k}\left(X^{k-1}+\sum_{1\le |r|\le X^{k-1}}d_{k-1}(|r|)\min\left\{X,\|D r\alpha_k\|_{\mathbb R/\mathbb Z}^{-1}\right\}\right), \end{align*} where the $(k-1)$-fold divisor function is the map \begin{align*} d_{k-1}:\mathbb N\to\mathbb N,\qquad m\mapsto \#\{(u_1,\dots,u_{k-1})\in\mathbb N^{k-1}:u_1\cdots u_{k-1}=m\} \end{align*}. We use the elementary divisor bound: for each $\eta>0$ there is a constant $C_{k,\eta}>0$ such that \begin{align*} d_{k-1}(m)\le C_{k,\eta}m^\eta \end{align*} for every $m\in\mathbb N$. Indeed, this follows by induction on $k-1$ from the estimate $\#\{d\in\mathbb N:d\mid m\}\le 2m^{\eta/2}$ for all sufficiently large $m$, with the finitely many smaller values absorbed into $C_{k,\eta}$. Since $1\le r\le X^{k-1}$ implies $r^\eta\le X^{(k-1)\eta}$, choosing $\eta>0$ so that $(k-1)\eta\le\varepsilon$ absorbs the divisor factor into $X^\varepsilon$. Hence \begin{align*} |S|^{2^{k-1}}\lesssim_{k,\varepsilon} X^{2^{k-1}-k+\varepsilon}\left(X^{k-1}+\sum_{1\le r\le X^{k-1}}\min\left\{X,\|D r\alpha_k\|_{\mathbb R/\mathbb Z}^{-1}\right\}\right). \end{align*} [/step]

[step:Estimate the averaged reciprocal distances using the rational approximation]We use the following Vinogradov rational-approximation summation estimate with a bounded approximation constant: if $Y\ge 1$, $R\ge 1$, $A\ge 1$, $\theta\in\mathbb R$, and there are coprime integers $b$ and $Q\ge 1$ with \begin{align*} \left|\theta-\frac{b}{Q}\right|\le A Q^{-2}, \end{align*} then \begin{align*} \sum_{1\le r\le R}\min\{Y,\|r\theta\|_{\mathbb R/\mathbb Z}^{-1}\}\lesssim_{A,\eta} (RYQ^{-1}+R+Q)Y^\eta \end{align*} for every $\eta>0$. This is the standard block decomposition form of Vinogradov's summation estimate; the bounded constant $A$ is part of the hypothesis and is carried into the implicit constant. Apply the lemma to $\theta=D\alpha_k$. Reduce $Da/q$ to lowest terms, writing $Da/q=b/Q$ with $\gcd(b,Q)=1$. Then $Q=q/g$ for some divisor $g$ of $D$, so $q/D\le Q\le q$. Moreover \begin{align*} \left|D\alpha_k-\frac{b}{Q}\right|=D\left|\alpha_k-\frac{a}{q}\right|\le Dq^{-2}\le DQ^{-2}. \end{align*} Thus the lemma applies with $A=D$, and all constants depend on $D=k!$, hence only on $k$. With $R:=\lfloor X^{k-1}\rfloor$ and $Y:=X$, the lemma gives \begin{align*} \sum_{1\le r\le X^{k-1}}\min\left\{X,\|D r\alpha_k\|_{\mathbb R/\mathbb Z}^{-1}\right\}\lesssim_{k,\varepsilon}X^\varepsilon\left(X^kq^{-1}+X^{k-1}+q\right). \end{align*} Since \begin{align*} X^kq^{-1}+X^{k-1}+q=X^k(q^{-1}+X^{-1}+qX^{-k}), \end{align*} and since $q\le X^k$ implies $X^{k-1}\le X^k(q^{-1}+X^{-1}+qX^{-k})$, we conclude that \begin{align*} X^{k-1}+\sum_{1\le r\le X^{k-1}}\min\left\{X,\|D r\alpha_k\|_{\mathbb R/\mathbb Z}^{-1}\right\}\lesssim_{k,\varepsilon}X^{k+\varepsilon}(q^{-1}+X^{-1}+qX^{-k}). \end{align*}[/step]

[guided]The point of this step is to convert the Diophantine information on $\alpha_k$ into an average bound for the reciprocal distances $\|Dr\alpha_k\|_{\mathbb R/\mathbb Z}^{-1}$. The correct summation input must allow a bounded constant in the rational approximation: after multiplying $\alpha_k$ by $D=k!$, the approximation error is multiplied by $D$. We use the Vinogradov rational-approximation summation lemma in the following form. If $Y\ge 1$, $R\ge 1$, $A\ge 1$, $\theta\in\mathbb R$, and $\theta$ has a reduced rational approximation $b/Q$ satisfying \begin{align*} \left|\theta-\frac{b}{Q}\right|\le A Q^{-2}, \end{align*} then, for every $\eta>0$, \begin{align*} \sum_{1\le r\le R}\min\{Y,\|r\theta\|_{\mathbb R/\mathbb Z}^{-1}\}\lesssim_{A,\eta}(RYQ^{-1}+R+Q)Y^\eta. \end{align*} This estimate is proved by grouping $r$ into blocks of length $Q$ and using the spacing of the residues $rb\bmod Q$; the constant depends on $A$ because the approximation is allowed to be within $AQ^{-2}$ rather than exactly $Q^{-2}$. Now set $\theta:=D\alpha_k$. Reduce $Da/q$ to lowest terms and write $Da/q=b/Q$ with $\gcd(b,Q)=1$. Since the reduction cancels a divisor of $D$, one has $Q=q/g$ for some $g\mid D$, hence $q/D\le Q\le q$. The approximation hypothesis gives \begin{align*} \left|D\alpha_k-\frac{b}{Q}\right|=D\left|\alpha_k-\frac{a}{q}\right|\le Dq^{-2}\le DQ^{-2}. \end{align*} Therefore the lemma applies with $A=D$, and its implicit constant depends only on $k$ and $\eta$. Taking $R:=\lfloor X^{k-1}\rfloor$ and $Y:=X$, and using $Q\asymp_k q$, gives \begin{align*} \sum_{1\le r\le X^{k-1}}\min\left\{X,\|D r\alpha_k\|_{\mathbb R/\mathbb Z}^{-1}\right\}\lesssim_{k,\varepsilon}X^\varepsilon\left(X^kq^{-1}+X^{k-1}+q\right). \end{align*} Finally, \begin{align*} X^kq^{-1}+X^{k-1}+q=X^k(q^{-1}+X^{-1}+qX^{-k}). \end{align*} The extra $X^{k-1}$ term already present outside the sum is also bounded by $X^k(q^{-1}+X^{-1}+qX^{-k})$, because the middle summand is exactly $X^{k-1}$. Hence \begin{align*} X^{k-1}+\sum_{1\le r\le X^{k-1}}\min\left\{X,\|D r\alpha_k\|_{\mathbb R/\mathbb Z}^{-1}\right\}\lesssim_{k,\varepsilon}X^{k+\varepsilon}(q^{-1}+X^{-1}+qX^{-k}). \end{align*}[/guided]

[step:Insert the summation estimate and take the final root]Substituting the preceding estimate into the differenced inequality yields \begin{align*} |S|^{2^{k-1}}\lesssim_{k,\varepsilon}X^{2^{k-1}+\varepsilon}(q^{-1}+X^{-1}+qX^{-k}). \end{align*} Taking the $2^{k-1}$-st root gives \begin{align*} |S|\lesssim_{k,\varepsilon}X^{1+\varepsilon}\left(q^{-1}+X^{-1}+qX^{-k}\right)^{2^{1-k}}, \end{align*} after replacing $\varepsilon$ by a smaller positive constant depending only on $k$. Since $S=\sum_{\substack{n\in\mathbb Z, 1\le n\le X}}e(P(n))$, this is exactly the desired estimate.[/step]

[guided]The previous steps give the powered estimate \begin{align*} |S|^{2^{k-1}}\lesssim_{k,\varepsilon}X^{2^{k-1}+\varepsilon}(q^{-1}+X^{-1}+qX^{-k}). \end{align*} Taking the $2^{k-1}$-st root is legitimate because both sides are nonnegative [real numbers](/page/Real%20Numbers). This gives \begin{align*} |S|\lesssim_{k,\varepsilon}X^{1+\varepsilon/2^{k-1}}\left(q^{-1}+X^{-1}+qX^{-k}\right)^{2^{1-k}}. \end{align*} Since the theorem asks for the estimate for every positive $\varepsilon$, we replace the earlier exponent parameter by $2^{k-1}\varepsilon$ and obtain \begin{align*} |S|\lesssim_{k,\varepsilon}X^{1+\varepsilon}\left(q^{-1}+X^{-1}+qX^{-k}\right)^{2^{1-k}}. \end{align*} By definition, $N=\lfloor X\rfloor$, so the integers $n$ with $1\le n\le X$ are exactly $1,\dots,N$. Therefore \begin{align*} S=\sum_{\substack{n\in\mathbb Z, 1\le n\le X}}e(P(n)), \end{align*} and the displayed bound is precisely the claimed estimate.[/guided]