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Statement

Here $C_\infty$ is the unique infinite component of $\mathcal R=\{(m,n)\in\mathbf Z^2:\gcd(m,n)=1\}$ under distance-1 adjacency (Proposition 3.1).

Lemma 7.1 (p. 58). The following sets lie in $C_\infty$:

1. $\{(m,1):m>0\}$;
2. $\{(p,n):p>n\}$ for each prime $p$;
3. $\{(m,q):q<m<(q/2)^{20/11},\ q\nmid m\}$ for each prime $q$.

The sets are printed as above, with no lower bound on $n$ in the second. The proof refers that set to the proof of Proposition 3.1, which treats $\{(p,n):1\le n\le p-1\}$.

Proof pointer

p. 58. Parts 1 and 2 are the observations in the proof of Proposition 3.1 (p. 50). For part 3, the range gives $q>2m^{11/20}$. So if $q\nmid m$, one of the intervals $[m,m+m^{11/20}]$ and $[m-m^{11/20},m]$ contains no multiple of $q$, and the horizontal segment at height $q$ over it lies in $\mathcal R$. By the Heath-Brown--Iwaniec theorem that every interval of length $y^{11/20}$ contains a prime, that segment crosses a prime column, which is in $C_\infty$ by part 2.

Read depth

Claims checked: the statement and proof were read clause by clause on p. 58 of the edition named on the source card. Nothing here is independently reviewed.

Dependencies

• Proposition 3.1 (p. 50), for parts 1 and 2.
• D. R. Heath-Brown and H. Iwaniec, On the difference between consecutive primes, Invent. Math. 55 (1979), 49--69, the paper's reference [34], for primes in every interval of length $y^{11/20}$.

Source. Ilan Vardi, "Deterministic Percolation," Communications in Mathematical Physics 207 (1999), 43--66, DOI 10.1007/s002200050717, the edition read for the source card.

Bears on

• Problem 1212: the lemma places in the infinite component of the problem's graph, taken over all of $\mathbf Z^2$, a line with a coordinate $1$, segments with a prime first coordinate and segments with a prime second coordinate. The paper does not consider paths that avoid coordinate $1$ or pairs of primes, so the lemma does not address the problem's question.