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Statement

Setting (p. 6, display (1.4)). For $V\subseteq\mathbb U$, the angle is $\mathrm{Ang}(V)=\sup_{v\in V,\,v\ne1}\lvert\arg(1-v)\rvert$, so $0\le\mathrm{Ang}(V)\le\pi/2$, with $\mathrm{Ang}({1})=\mathrm{Ang} (\emptyset)=0$. $\mathbb T$ is the unit circle, and $\Gamma(S)$ is the spectrum of the Theorem 1 page.

Theorem 5 (p. 9, quoted). "Suppose $S$ is a closed subset of $\mathbb U$ with $1\in S$. The spectrum of $S$ is $\mathbb U$ if and only if $\mathrm{Ang}(S)=\pi/2$. If $\mathrm{Ang}(S)=\theta<\pi/2$, then there exists a positive constant $A(\theta)$, depending only on $\theta$, such that $\Gamma(S)$ is contained in a disc centered at $A(\theta)$ with radius $1-A(\theta)$. In fact, $A(\theta)=(28/411)\cos^2\theta$ is permissible. Thus

$$\Gamma(S)\cap\mathbb T=\begin{cases}\{1\}&\text{if }Ang(S)<\pi/2\\ \mathbb T&\text{if }Ang(S)=\pi/2.\end{cases}"$$

For $S=[-1,1]$, where $\theta=0$, the disc is centred at $28/411$ with radius $383/411$, so $\Gamma([-1,1])\subset[-355/411,1]$; the paper records (p. 9) that this gives some $c>-1$ with $\Gamma(S)\subset[c,1]$ and so generalizes Hall's theorem on Heath-Brown's conjecture. The exact value is $\delta_1=-0.656999\ldots$ by Theorem 1. The paper also notes (p. 10) that $A(\theta)\le(1-\exp(-\pi\cot\theta))/2 \le\frac\pi2\cos\theta$, so $A(\theta)$ must tend to $0$ as $\theta\to\pi/2$.

Source. Andrew Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. (2) 153 (2001), no. 2, 407--470; read as arXiv:math/9909190v1 (8 September 1999), printed page $=$ PDF page: the angle on p. 6, Theorem 5 on p. 9, the remark on p. 10, Sections 7c and 7d on pp. 45--49. The published pagination differs and was not compared. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the remarks after it were read clause by clause on the page images. The proof was not checked; the arithmetic $2\cdot28/411-1=-355/411$ is this page's.

Proof pointer

Sections 7c and 7d (pp. 45--49). With $\lambda=(28/411)\cos^2\theta$ the paper shows that every value $\sigma(u)$ of a solution of (1.5) lies within $1-\lambda$ of $\lambda$. With $P(u)=\int_0^u\min(2,(1-\mathrm{Re}\,\chi(t))\sec^2\theta)\,dt/t$ and $u_0$ the point where $P(u_0)+P(u_0/2)=1$ ($u_0=\infty$ if there is none, p. 45), Proposition 7.5 bounds $\lvert\sigma(u)\rvert$ by $1-2\lambda$ for $u\ge u_0$; for $u\le u_0$, the inequality (7.4), the second inequality of (7.6) and (7.7) give $(\mathrm{Re},\sigma(u)-\lambda)^2+(\mathrm{Im}, \sigma(u))^2\le(1-\lambda)^2$ (p. 49). So $\Lambda(S)$ lies in the disc, and Theorem 5 follows from $\Gamma(S)\subset[0,1]\times\Lambda(S)$ (Theorem 3$'$, p. 9).

Dependencies

Theorem 3$'$ (p. 9), Lemma 7.2 with (7.4) and (7.6) (p. 46), the inequality (7.7) (p. 47) and Proposition 7.5 (p. 48) of Section 7c of the same paper; see also Theorem 3.

Bears on

No Erdős problem page of the corpus cites this theorem.