Exact indicator extremizers for complete-grid large sieve inequalities

Definitions and mathematical statement

Write $e(t)=e^{2\pi it}$ and $S(\alpha)=\sum_{n=M+1}^{M+N}a_ne(\alpha n)$, with integer $M$, $N\ge1$. Helfgott's source question asks whether the standard large sieve remains best possible when $a_n\in\{0,1\}$; Vaughan's affirmative response is a prediction. The excerpt does not specify the sampling set, exact constant or asymptotic regime, and the supplied record distinguishes two interpretations.

In the analytic version, points $\alpha_j\in{\mathbb{R}}/{\mathbb{Z}}$ are circularly $\delta$-separated, and \[ \sum_j|S(\alpha_j)|^2\le(N-1+\delta^{-1})\sum_n|a_n|^2. \] The report's complete-grid lemma takes $\alpha_j=j/R$, $0\le j<R$, $R\ge1$, and asserts \[ \sup_{0\ne a\in\{0,1\}^N} \frac{\sum_{j=0}^{R-1}|S(j/R)|^2}{\sum_n a_n^2} =R\lceil N/R\rceil. \] This equals the unrestricted complex-coefficient operator norm. An extremizer is the indicator of a residue class modulo $R$ containing the largest number of indices in the length-$N$ interval. If $N\equiv1\pmod R$, the value is exactly $N+R-1$. For fixed $0<\delta\le1$, choose $R=\lfloor\delta^{-1}\rfloor$, so the grid is admissibly separated; along infinitely many such lengths the gap from $N-1+\delta^{-1}$ is $\delta^{-1}-R<1$. Exact equality with that analytic factor occurs when $\delta^{-1}=R$.

In the distinct arithmetic/Farey version, the sampling set is all reduced fractions of denominator at most $Q$, including $1/1$ as zero modulo one: \[ \mathcal L_Q(a)=\sum_{q\le Q}\sum_{\substack{1\le h\le q\$h,q)=1}}|S(h/q)|^2 \le(N-1+Q^2)\sum_n|a_n|^2. \] The supplied verification obtains this from the analytic spacing $\delta=Q^{-2}$. The report says singleton and all-ones indicators show the $N+Q^2$ order up to an absolute factor, but leave its critical leading constant unresolved. A complete rational grid is not silently identified with this fixed Farey set. Thus the exact indicator-extremizer result resolves the grid/analytic sharpness interpretation, not every possible arithmetic leading-constant formulation of the source question.

Short English statement

Is the large sieve still sharp for zero-one coefficients? The supplied report gives exact complete-grid indicator extremizers and near-sharp analytic-spacing constants, while distinguishing the Farey-sampling problem where only order-of-magnitude sharpness is claimed.

Sources

• Local UnsolvedMath registry, permanent ID $20000740$, source code catalogue entry; AIM analytic number theory corpus, aim-analytic-number-theory-notes.json, notes[103]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM Gaps between primes, item 2, "On the large sieve, I", question by H. Helfgott and comment by R. C. Vaughan: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf. Original remarks and literature fields are empty. The extracted clean field is damaged; the original exponential-sum target is retained.
• The supplied verification explicitly treats both analytic separated-point and arithmetic/Farey readings, whose sharpness questions differ. No independent source inspection or complete primary large-sieve citation is supplied here.
• Registry dated August 14, 2026: statement reconstructed_unverified, status partially_solved, classification PARTIAL-PROGRESS; attempt 1 records problem status unknown, partial result, rigor proved, novelty confidence low. The grid norm and restricted sharpness conclusions are machine-generated research aids, not peer reviewed, requiring independent verification before citation.