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Let $\alpha$ be a badly approximable number, and sum $B={\sum }_{D<d\le 2D}{\sum }_{W<w\le 2W,dw\le X}{a}_d{b}_{w}e(\alpha dw)$ with $|a_d|\le \tau (d)$ and $|b_w|\le 1$. The Cauchy-Schwarz inequality, divisor-square summatory bound and geometric sums give $|B{|}^2\ll D{\log }^{3}X(DW+W{\sum }_{h\le W}\min (D,\Vert h\alpha {\Vert }^{-1}))$. Separated rotations bound the last sum by $O_{\alpha }(W\log (2W))$, giving the displayed estimate. A hyperbolic cutoff remains valid because the allowed first-variable values after expansion form an interval.

For a positive nonsquare integer $D$ and positive integers $a,q$, the integer $D{q}^2-a^2$ is nonzero. Therefore $|\sqrt{D}-a/q|=|D{q}^2-a^2|/(q^2(\sqrt{D}+a/q))$. If $a/q\le \sqrt{D}+1$, the denominator factor is at most $2\sqrt{D}+1$; otherwise the approximation error already exceeds one. This proves that a quadratic irrational of this form is a badly approximable number. For $D=2$, splitting instead at $a/q=2$ gives the strict bound $|\sqrt{2}-a/q|>1/(4q^2)$.

// Target: analytic-number-theory.bigb

Use the integer norm lower bound for rational approximation to a quadratic irrational. The integer $2q^2-a^2$ is nonzero, since $\sqrt{2}$ is irrational. If $a/q<2$, thenbecause $\sqrt{2}+a/q<\sqrt{2}+2<4$. If $a/q\ge 2$, the difference is at least $2-\sqrt{2}>1/4\ge 1/(4q^2)$. ThusIn particular, $\sqrt{2}$ is a badly approximable number.

To estimate the prime-weighted exponential sum, let $Q=\lfloor N^{1/2}\rfloor$. The Dirichlet approximation theorem gives a reduced fraction $a/q$ withCombining this with the lower bound gives $q>Q/4$. Thus $q\asymp N^{1/2}$; the approximation cannot have a small denominator.

The relevant analytic input is the prime exponential-sum estimate from Vaughan identity:Here is the structure behind that estimate. With Dirichlet convolution and cutoffs $U,V$, the Vaughan identity isIt follows from $\Lambda =\mu \ast \log$, $\log =\Lambda \ast 1$, and $\mu \ast 1={\delta }_1$. The first two Dirichlet convolution terms become type I sums with one short factor; the last becomes a bilinear type II sum with both factors bounded below. Split these sums into dyadic blocks and use $U=V=N^{2/5}$. For the type I sums, the inner finite geometric series has bound $\min (L,(2\Vert \alpha m\Vert {)}^{-1})$. For the type II sums, the Cauchy-Schwarz inequality followed by expansion introduces analogous sums in differences of indices. Rational approximation separates the phases into blocks of length comparable to $q$, bounding these reciprocal-distance sums. The Dirichlet convolution coefficients are divisor-bounded, producing the logarithmic factors. These steps give the displayed estimate; no unproved cancellation of arbitrary weighted sums is being assumed.

With $\alpha =\sqrt{2}$ and $q\asymp N^{1/2}$, the two denominator-dependent terms are $O(N^{3/4})$. ConsequentlyThe final passage absorbs the logarithm using the positive exponent margin $9/10-4/5=1/10$.

For reduced $a/q$ with $|\alpha -a/q|\le q^{-2}$ and $1\le q\le N$, the displayed bound follows by splitting Vaughan identity with parameters $U=V=N^{2/5}$. Type I sums use short geometric-sum bounds; type II sums use the Cauchy-Schwarz inequality and rational spacing to control bilinear cancellation. A badly approximable number has a Dirichlet approximant with $q\asymp N^{1/2}$, yielding $O(N^{4/5}{\log }^{4}N)$ and hence $O(N^{9/10})$.

// Target: analytic-number-theory.bigb