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For every $0\le \beta\le 1/2$, we construct a nonzero real-valued continuous function $f_\beta$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat f_\beta=f_\beta$ and $$ f_\beta\!\left(\frac{\sqrt n}{[\log(e+n)]^\beta}\right)=0 $$ for all $n\ge0$. The case $\beta=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula.
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The authors state that OpenAI's ChatGPT provided exploratory input and calculations that were essential in the development of the paper. The disclosure does not identify a specific model version or isolate which theorem, lemma, or proof step originated from ChatGPT, so the safest attribution is substantive AI-assisted discovery rather than AI-discovered.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Unreviewed. arXiv 2608.13468 read here; the acknowledgement thanks ChatGPT, "whose exploratory input and calculations were essential in the development of this paper", naming no model version and isolating no step. A complete conventional proof by the authors; not refereed; no formalization.
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For every $0\le \beta\le \tfrac12$, Bondarenko and Seip construct a nonzero real-valued continuous even function $$ f_\beta\in L^1(\mathbb R)\cap L^2(\mathbb R) $$ such that $$ \widehat f_\beta=f_\beta $$ and $$ f_\beta\!\left(\frac{\sqrt n}{[\log(e+n)]^\beta}\right)=0 \qquad(n\ge0). $$ They normalize the construction by requiring $f_\beta(1/2)=1$, so the function is genuinely nontrivial. For $\beta=0$, this gives a nonzero Fourier-invariant function vanishing at every $\sqrt n$, which answers the question negatively: their interpolation formula for even Schwartz functions does not extend merely under the assumption that the interpolation series is well-defined and absolutely convergent. More strongly, for every $0<\beta\le1/2$ the zero set can be logarithmically denser than the square-root sequence. These sampling points, together with $1/2$, form a universal interpolating sequence for a suitable reproducing-kernel Hilbert space of Fourier-invariant Hermite expansions.
Source-reported tools: argument.
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