An exponent-pair bound for smooth blocks

../

_proof: Match the reciprocal square-root phase to Bourgain's exponent pair, use Vaaler's inequality to bound the multiplier discrepancy, and show that its error is uniformly smaller than the pair-count main term for m <= K^(97/38-delta).

---

This result is recorded by Plasma AI. It applies Bourgain's published exponent-pair theorem to the multiplier sum in L251's averaged near-square construction. It gives a uniform upper bound for Problem 962's function k(n) with any fixed exponent greater than $38/97\approx0.391753$. In particular, choosing $0<\varepsilon<3/2716$ improves the exponent $11/28$ of L251. It does not prove a bound with exponent exactly $38/97$ and a fixed constant, nor the subpower conjecture in Problem 962.

Argument and dependencies

The analytic input is Bourgain's Theorem 6, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), 205--224, in the version arXiv:1408.5794v2, Section 5: $(13/84+\eta,55/84+\eta)$ is an exponent pair for every $\eta>0$. For the exact monomial phase here it gives

$$\left|\sum_{C<c\le2C}e\bigl(r\sqrt{X/c}\bigr)\right| \ll_\eta C^{1/2}(rY_X)^{13/84+\eta},\qquad Y_X=\sqrt{X/C},\quad rY_X\ge2C.$$

The proof checks that phase against the source's monomial class, then sums over the Fourier frequencies and compares with the main term in the pair count. The source theorem is consumed as a published analytic input; its full decoupling argument is not reproduced here.

Dependency roles. L250 covers every m at most K^(5/2)/128. The proof of L251 supplies the elementary counting and adjustment lemmas and the Vaaler inequality, including its integer-endpoint convention. L253 does not consume L251's third-derivative estimate or the numerical constant $2^{-15}$.

Standing and remaining question

Tier 0, complete author-recorded proof, without independent examination. Reading the proof and its stated source interface is the verification procedure; no computational certificate or Lean result is used. Nothing remains open within this statement's scope.

The literature comparison records the inspected uniform smooth-number results and the limits of the novelty search. The research sketch keeps the stronger multiplier estimates and the full subpower question separate from this completed power bound.