bibkey: gafnitao2026exceptionalintervals authors: Ayla Gafni; Terence Tao year: 2026 title: On the number of exceptional intervals to the prime number theorem in short intervals doi: 10.2140/ent.2026.5.221 url: https://arxiv.org/abs/2505.24017v1 claim: Theorem 1.2 bounds the measure exponent of fixed-relative-error short Chebyshev increments through zero density; it does not identify Robin excess states with that exceptional set or provide their prime-sampled count. strata_touched: [] license: citation-only triage: anchor

Short-increment exceptional sets and the Robin baseline

The inspected primary is the versioned HTML of arXiv:2505.24017v1, submitted 29 May 2025. Its abstract metadata lists Essential Number Theory 5 (2026), 221–241 and the DOI above. The journal edition, complete proof and numerical optimizations were not independently audited. The definitions and theorem locators below refer to that manuscript. No Lean certification is claimed.

The averaging variable and fixed tolerance

Definition 1.1 fixes $0<\theta <1$, $X>1$ and $\epsilon >0$, and defines a set of real starting points

$${\mathcal{E}}_{\epsilon }(X,\theta )=\{x\in [X,2X]:|\psi (x+x^{\theta })-\psi (x)-x^{\theta }|\ge \epsilon x^{\theta }\}.$$

Here $\psi$ includes every prime power. Let ${\mu }_{\epsilon }(\theta )$ be the infimum of the exponents $\xi$ for which its Lebesgue measure is ${\ll }_{\epsilon ,\theta }{X}^{\xi }$ eventually, and set $\mu (\theta )={\sup }_{\epsilon >0}{\mu }_{\epsilon }(\theta )$. An eventually empty set has exponent $-\infty$. The tolerance is fixed; no uniform rate as $\epsilon \downarrow 0$ is supplied by this definition.

Section 1.1 defines $A(\sigma )$ by $N(\sigma ,T)\le T^{A(\sigma )(1-\sigma )+o(1)}$ at fixed $0\le \sigma <1$, counting zero multiplicities and both ordinate signs. It sets $A(\sigma )=-\infty$ if that zero set is empty. Theorem 1.2 gives

$$\mu (\theta )\le \underset{ϵ>0}{\inf }\underset{\begin{array}{c}0\le \sigma <1\\ A(\sigma )\ge (1-\theta {)}^{-1}-ϵ\end{array}}{\sup }[(1-\theta )(1-\sigma )A(\sigma )+2\sigma -1].$$

The empty supremum is $-\infty$. The small $ϵ$ in this formula is distinct from the fixed exceptional-set tolerance. The source explicitly keeps the infimum because continuity of the actual density exponent is not known. Theorem 1.3 refines the right side by also using the source’s zero additive-energy exponent; neither theorem asserts a Robin-excess inclusion or a prime-sampling law. The published density and energy arguments are reused rather than recomputed here.

Normal short increments need not fund a Robin baseline

The existing uniform short-interval input already gives $\psi (x+x^{2/3})-\psi (x)=x^{2/3}(1+o(1))$ uniformly at large $x$, including its paid prime-power correction. Thus ${\mathcal{E}}_{\epsilon }(X,2/3)$ is empty eventually for each fixed $\epsilon >0$. This is a direct instance of that existing input, not a new exceptional-set calculation.

The same-source persistence application uses normal short-interval increments to bound variation while preserving a potentially positive Robin baseline. Under its RH-failure hypothesis, the selected excess persists despite these normal increments. No implication from an excessive full CA Robin state to ${\mathcal{E}}_{\epsilon }(X,2/3)$ has been established. Such an implication would supply the missing exclusion itself.

At shorter lengths, the general measure theorem still concerns fixed relative errors in increments. An estimate on that set cannot be transported to a count of data-dependent first-prime CA samples without a proved inclusion and sampling comparison. A shrinking tolerance, the exponent tail and the actual $\log C_b$ denominator also retain separate obligations. No one-sided prime-prefix count, complete signed Robin tail estimate or RH proof is obtained.