A quantitative short-interval transfer for small values of ap-q

Definitions and mathematical statement

Problem 14 (A. Balog). For fixed integer $a>1$, show that $|ap-q|$ can be small with $p,q$ prime. More generally consider real $a$. The final OCR fragment is "$a,1$", with its comparison symbol lost; it may intend an exclusion of $a=1$, but this is not silently restored as exact text. "Small" has no quantitative scale or infinitude/density requirement in the source.

Reported quantitative short-interval transfer. For every fixed $a>0$ and every $\eta>0$, the aid asserts that there exist $c(a,\eta)>0$, $C(a,\eta)>0$, and $P_0(a,\eta)$ such that, for every $P\geq P_0$, at least \[ c(a,\eta)\frac P{\log P} \] primes $p\in(P,3P/2)$ have a prime $q$ with \[ 0<q-ap<C(a,\eta)p^{1/20+\eta}. \] This is one-sided, excludes the zero difference, holds block by block, and implies a sublinear error by choosing $\eta<19/20$. It is not a bounded-error result. The report derives it from Jia's published quantitative almost-all short-interval theorem with exceptional set $O(X(\log X)^{-B})$ and a bounded-multiplicity endpoint map. The exact theorem statement, the role/range of $B$ and the complete citation are absent.

The candidate-contribution summary also gives an ordered-prime-pair lower bound of order \[ \frac{P^{1+1/20+\delta}}{(\log P)^2} \] at error scale $p^{1/20+\delta}$, for the designated $a>0$. Its precise $\delta$ range, interval endpoints and constants are not supplied, so it is retained as a qualified report rather than an unqualified theorem for all $\delta$.

Reported rational local optimum. Let $a=A/B>0$ in lowest terms, $A,B\in{\mathbb{N}}$. For odd prime inputs and outputs, \[ B(q-ap)=Bq-Ap\in{\mathbb{Z}}, \] which is odd if $AB$ is even and even if $AB$ is odd. Thus the smallest arithmetically permitted strictly positive error is \[ \frac{d_{A,B}}B,\qquad d_{A,B}=\begin{cases}1,&AB\text{ even},\\2,&AB\text{ odd}.\end{cases} \] The aid's local calculation identifies no further obstruction and, conditional on Dickson's conjecture for the corresponding admissible pair of linear forms, predicts infinitely many prime pairs satisfying \[ Bq-Ap=d_{A,B}. \] Consequently it calls $1/B$ or $2/B$ the conjectural optimal bounded error. Exceptional pairs involving the prime $2$ do not affect this infinite-family formulation. For the absolute-value question one must also handle zero differences: if $a=1$ and $p=q$ is allowed, that reading has the trivial value zero. The strict one-sided transfer and rational optimum above avoid that degeneracy. The report does not establish Dickson's conjecture or the bounded-error prediction unconditionally.

Short English statement

Find primes with small $|ap-q|$ for a fixed slope. The aid reports a block-by-block positive proportion of prime inputs with one-sided error $p^{1/20+\eta}$. For rational slopes it separately identifies the conjectural smallest positive bounded error from parity and denominator restrictions, with infinite attainability conditional on Dickson.

Sources

• Local UnsolvedMath registry, permanent ID $20000749$, catalogue entry; canonical aim-analytic-number-theory-notes.json, notes[112].
• AIM workshop "Gaps between primes," PDF item 14, proposed by A. Balog: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf. Supplied verification repeats OCR wording; the final real-$a$ qualification is damaged.
• Jia's quantitative exceptional-set short-interval theorem and Dickson's conjecture, as named in the supplied summary. No full titles, identifiers or theorem statements are supplied.
• Research attempt 1, partial_result, run status partially_solved, internal rigor proved, low novelty confidence. Explicitly a machine-generated research aid, not peer reviewed; independently verify before citation. Original literature field empty.