{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets", "title": "The Radchenko–Viazovska question on Fourier interpolation", "canonical_statement": "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\n$$\nf_\\beta\\!\\left(\\frac{\\sqrt n}{[\\log(e+n)]^\\beta}\\right)=0\n$$\nfor all $n\\ge0$. The case $\\beta=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula.", "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "resolved", "solution_events": [ { "id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets:event", "problem_id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets", "type": "disproved", "status": "resolved", "occurred_at": "2026-08-13T00:00:00.000Z", "title": "arXiv", "summary": "For every $0\\le \\beta\\le \\tfrac12$, Bondarenko and Seip construct a nonzero real-valued continuous even function\n$$\nf_\\beta\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\n$$\nsuch that\n$$\n\\widehat f_\\beta=f_\\beta\n$$\nand\n$$\nf_\\beta\\!\\left(\\frac{\\sqrt n}{[\\log(e+n)]^\\beta}\\right)=0\n\\qquad(n\\ge0).\n$$\nThey normalize the construction by requiring $f_\\beta(1/2)=1$, so the function is genuinely nontrivial.\n\nFor $\\beta=0$, 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.\n\nMore 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.", "ai_contribution": "ai_assisted", "attempt_ids": [ "vibemathed:fourier-invariant-functions-with-dense-zero-sets:attempt" ], "method_family_ids": [ "vibemathed:fourier-invariant-functions-with-dense-zero-sets:method" ], "source_assertion_ids": [ "vibemathed:fourier-invariant-functions-with-dense-zero-sets:assertion" ] } ], "attempts": [ { "id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets:attempt", "problem_id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets", "model": "ChatGPT (OpenAI, model version unstated)", "provider": "OpenAI", "attempted_at": "2026-08-13T00:00:00.000Z", "human_collaborators": [ "Andriy Bondarenko", "Kristian Seip" ], "exposure": "unknown", "prompt_public": null, "outcome": "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.", "failed_routes": [], "artifacts": [], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.13468", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The Radchenko–Viazovska question on Fourier interpolation", "url": "https://vibemathed.com/problem/fourier-invariant-functions-with-dense-zero-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets:verification", "problem_id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets", "solution_event_id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "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.", "sources": [ { "label": "VibeMathed: The Radchenko–Viazovska question on Fourier interpolation", "url": "https://vibemathed.com/problem/fourier-invariant-functions-with-dense-zero-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "arXiv", "url": "https://arxiv.org/abs/2608.13468", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.13468", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The Radchenko–Viazovska question on Fourier interpolation", "url": "https://vibemathed.com/problem/fourier-invariant-functions-with-dense-zero-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets:signal:watch_only", "problem_id": "vibemathed:fourier-invariant-functions-with-dense-zero-sets", "kind": "watch_only", "reason": "Current public evidence does not meet the default replay threshold.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }