Determine the infimum and supremum of$$|\{x\in \mathbb{R}:|f(x)|<1\}|$$as $f\in \mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are real and in the interval $[-1,1]$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let [ S_f:={x\in\mathbb R:\ |f(x)|<1},\qquad m(f):=|S_f| ] (Lebesgue measure). Note that replacing “(<1)” by “(\le 1)” does **not** change the measure, because (|f(x)|=1) has only finitely many real solutions.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.