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This proves the whole statement of L253. Write $e(u)=\exp(2\pi iu)$, $\psi(u)=u-\lfloor u\rfloor-\frac12$, and

$$S_C(X)=\sum_{C<c\le2C}\psi\bigl(\sqrt{X/c}\bigr),\qquad Y_X=\sqrt{X/C}.$$

All implied constants below are uniform in $m,K,C,X,r$. They may depend on the fixed positive parameter $\eta$, and eventually on $\delta$.

The analytic input and its specialization

Theorem 6 of Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), 205--224, in arXiv:1408.5794v2, Section 5, states that

$$(\kappa,\lambda)=(13/84+\eta,55/84+\eta)$$

is an exponent pair for every fixed $0<\eta<1/100$. In the normalization used there, an admissible phase of size $T$ on an interval of length comparable to $M$, with $T\ge M$, has exponential sum $\ll_\eta (T/M)^\kappa M^\lambda$. The phase class consists of functions whose derivatives approximate those of a monomial with first derivative $yu^{-s}$, $s>0$; the constants are uniform for fixed $s$ and fixed derivative tolerances. Section 5 explains both the extension to $M>\sqrt T$ and the allowance of subintervals. These qualifications matter here, since the first few frequencies can have $C>\sqrt{rY_X}$.

For $X>0$ and $r\ge1$, take

$$f(u)=-r\sqrt X\,u^{-1/2}\quad(C\le u\le2C).$$

Its first derivative is exactly $f'(u)=(r\sqrt X/2)u^{-3/2}$: this is the source's monomial with $s=3/2$, with zero approximation error at every derivative order. Equivalently, its normalized phase is $-v^{-1/2}$ on $1\le v\le2$, independent of $r,X,C$, and its size is $T=rY_X$. The positive-phase sum is the conjugate of the sum for $f$, so the same bound applies. Thus, whenever $rY_X\ge2C$,

$$T_r(X):=\sum_{C<c\le2C}e\bigl(r\sqrt{X/c}\bigr),\qquad |T_r(X)|\ll_\eta (rY_X/C)^\kappa C^\lambda =C^{1/2}(rY_X)^\kappa . \tag{1}$$

The equality uses $\lambda-\kappa=1/2$. The fixed change from the source's interval $[M/2,M]$ to $[C,2C]$ changes only constants. No almost-all qualification occurs in this estimate.

Discrepancy estimate

Assume $Y_X\ge2C$ and $C^{1/2}Y_X^{-\kappa}\ge1$. Put

$$V=(C^{1/2}Y_X^{-\kappa})^{1/(1+\kappa)},\qquad R=\lceil V\rceil,$$

so $V\le R\le2V$. Equation (1) of the proof of L251, derived from Vaaler's inequality for this exact convention of psi, gives

$$|S_C(X)|\le\frac{C}{2R+2} +\Bigl(1+\frac1\pi\Bigr)\sum_{r=1}^R\frac{|T_r(X)|}{r}.$$

Since $Y_X\ge2C$, (1) applies to every positive frequency, and $\sum_{r=1}^Rr^{\kappa-1}\ll_\eta R^\kappa$. Consequently

$$|S_C(X)|\ll_\eta \frac CR+C^{1/2}Y_X^\kappa R^\kappa \ll_\eta C^aY_X^b, \qquad a=\frac{\kappa+1/2}{1+\kappa},\quad b=\frac{\kappa}{1+\kappa}. \tag{2}$$

Both terms have exactly these exponents after substituting $V$. In particular

$$1+a=\frac{3/2+2\kappa}{1+\kappa},\qquad 1-b=\frac1{1+\kappa}. \tag{3}$$

Uniform construction

Fix $0<\delta<1/20$, set $\eta=\delta/10$ and $\nu=21/38-\delta$. Then $1/2<\nu<21/38$ and $0<\kappa<1/4$. All inequalities that follow hold for every $K$ beyond a single threshold depending only on $\delta$.

Let $K\ge64$ and $0\le m\le K^{2+\nu}$ be integers. If $m\le K^{5/2}/128$, apply L250's explicit construction. Otherwise put

$$C=\left\lceil\frac{16m}{K^2}\right\rceil,\qquad E=\left\lfloor\frac{K^2}{128C}\right\rfloor .$$

The following bounds are uniform throughout the remaining range:

$$\frac{\sqrt K}{8}<C\le32K^\nu=o(K),\qquad \frac{K^2}{256C}\le E\le\frac{K^2}{128C}\le m, \tag{4}$$

and, for $X=m$ and $X=m+E$,

$$\frac{K}{4\sqrt2}<Y_X\le\frac{K}{2\sqrt2}. \tag{5}$$

Indeed $16m/K^2>\sqrt K/8\ge1$, so $C\le32m/K^2\le32K^\nu$, while $C\ge16m/K^2$ bounds $Y_m$ above by $K/4$ and $C<16m/K^2+1\le32m/K^2$ bounds it below by $K/(4\sqrt2)$. For large $K$, $C\le K^2/256$, giving the lower bound for $E$ after rounding down. Also $C>\sqrt K/8$ gives $E<K^{3/2}/16\le m$ for $K\ge64$. This proves (4), and $m+E\le2m$ proves (5).

In particular, $Y_X\ge2C$ for all sufficiently large $K$, because $\nu<1$. Furthermore

$$C^{1/2}Y_X^{-\kappa} >\frac{K^{1/4-\kappa}}{\sqrt8}\longrightarrow\infty .$$

Thus (2) applies to both endpoints, giving

$$|S_C(m)|+|S_C(m+E)|\ll_\delta C^aK^b. \tag{6}$$

By Lemma 2 of the proof of L251, the number of pairs of positive integers $(c,q)$ with $C<c\le2C$ and $m<cq^2\le m+E$ is

$$\sum_{C<c\le2C} \left(\sqrt{(m+E)/c}-\sqrt{m/c}\right)-S_C(m+E)+S_C(m).$$

The elementary estimate (3) in that proof uses only (4), (5) and $Y_m\le K/4$, and gives the lower bound $K/(219C)$ for this sum's main term. To compare (6) with it, (3) and $C\le32K^\nu$ yield

$$\frac{C^aK^b}{K/C} \ll_\delta K^{\{\nu(3/2+2\kappa)-1\}/(1+\kappa)}. \tag{7}$$

Here the exponent is strictly negative. More precisely,

$$\nu(3/2+2\kappa)-1 =\left(\frac{21}{38}-\delta\right) \left(\frac{38}{21}+\frac\delta5\right)-1 =\delta\left(\frac{21}{190}-\frac{38}{21}\right) -\frac{\delta^2}{5}< -\delta .$$

It follows that the pair count is positive for every sufficiently large $K$, uniformly for all the integers $m$ in the specified range. Choose such a pair and let $t$ be the largest nonnegative integer satisfying $ct^2<cq^2-m$. The adjustment lemma in L251 gives

$$0<N-m<2\sqrt{cE}+c\le K/4+2C\le K, \qquad N=c(q-t)(q+t).$$

Also $1\le c\le2C\le K$, $q-t\ge1$, and

$$q\le\sqrt{(m+E)/c}\le\frac{K}{2\sqrt2},\qquad t<\sqrt{E/c}\le\frac{K}{8\sqrt2 C}.$$

Since $C\ge1$, these two bounds give $q+t<K$. All three factors are positive integers at most $K$. This proves the first sentence after choosing an integer $K_0(\delta)$ large enough for the finitely many uniform threshold requirements above. Since $2+\nu=97/38-\delta$, its exponent is exactly the one stated.

Translation to k(n)

Fix $\varepsilon>0$. Choose $0<\delta<1/20$ so small that $1/(97/38-\delta)<38/97+\varepsilon$, and write $d=97/38-\delta$. For every integer $n\ge1$, set

$$K=\max\bigl(K_0(\delta),\lceil n^{1/d}\rceil\bigr).$$

Then $n\le K^d$, and the first sentence supplies a $K$-smooth member of every block $(m,m+K]$ with $0\le m\le n$. If a longer block witnessed $k(n)\ge K$, its first $K$ terms would have no such member. Hence

$$k(n)\le K-1<\max\bigl(K_0(\delta),n^{1/d}\bigr) \le K_0(\delta)n^{38/97+\varepsilon}.$$

This gives the stated bound with $A_\varepsilon=K_0(\delta)$. Every construction included $m=0$, so requiring $m$ to be positive preserves all conclusions. $\square$