Averaging the multiplier pushes the smooth-block exponent from 2/5 to 11/28

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_proof: Count pairs (c, q) with C < c <= 2C and cq^2 in a window of length about K^2/C above m; Vaaler's inequality and the third-derivative test bound the error by about C^(4/7) K^(1/7), below the main term K/C when C is at most K^(6/11)/1024, and one adjustment lands within K of m.

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This result is recorded by Plasma AI. It improves the exponent of the proved upper bound for Problem 962, whose function k(n) it bounds by a constant times n^(11/28): $11/28\approx0.3929$, against $2/5$ from Erdős and Sárközy's 1979 construction and $1/2$ from the forum argument the problem page records. It does not touch the displayed question of Problem 962, which asks for $\log k(n)\le(\log n)^{1/2+o(1)}$.

Meaning of the statement

Erdős and Sárközy write a $K$-smooth member of the block as $c(q-t)(q+t)$ with one multiplier $c$ chosen from $m$ alone; that choice forces $m\ll K^{5/2}$, as L250 records. Here the multiplier ranges over a whole interval $C<c\le2C$, and one only needs some pair $(c,q)$ with $cq^2$ in a window $(m,m+E]$ of length $E\asymp K^2/C$; the adjustment by $t$ then lands within $K$ of $m$. The number of such pairs has main term of order $K/C$, and its error term is a sum of the sawtooth function $\psi$ at $\sqrt{X/c}$ over the multipliers. Vaaler's trigonometric approximation and the van der Corput third-derivative test bound that sum by $\ll C^{4/7}K^{1/7}+C^{3/4}(\log C)^{3/2}$, which stays below the main term for $C\le K^{6/11}/1024$, that is, for $m$ up to about $K^{28/11}$.

Standing and verification

Tier 0, author-recorded, with a complete proof; no independent review has examined it. Reading the proof is the verification procedure; every constant in it is explicit. The two analytic inputs are source results consumed as stated, with their hypotheses checked against the use in the proof: Vaaler's inequality and the explicit third-derivative test, as Lemmas 7.1 and 8.6 of Granville and Ramaré (1996), whose held statements supply both inputs print them. The proof records one misprint in the printed Lemma 7.1, the place of a sign, and shows that the inequality it uses holds for the corrected placement; only the moduli of the coefficients enter. Nothing remains open within the stated scope.

Dependency roles. L250 supplies the blocks with m at most K^(5/2)/128, where the averaging step is not needed; it is used at full strength, for every $K\ge64$.

Novelty and context

No every-interval result giving a diagonal exponent below $2/5$ was found in the literature search that the literature comparison for smooth integers in short intervals records; that search is bounded and is not a novelty certificate. By its abstract, Rivat and Shparlinski (2016) use the same toolkit, exponent pairs with Vaaler polynomials, for multiples of squares in intervals of length $x^{1/2}$, a different range; their full text was not read. The research sketch records the exponent this route gives from any exponent pair, the exact limits of that calculation and of a three-factor output, and the next tests toward Problem 962. The stronger asymptotic bound L253 uses Bourgain's exponent pair to obtain k(n) <<_epsilon n^(38/97+epsilon), while this claim retains an explicit bound for every $n\ge1$.