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Write $\omega(n)$ for the number of distinct prime divisors of $n$, with $\omega(1)=0$. The external input is Erdős–Pomerance–Sárközy Lemma 1: with modulus $m=1$, additive function $\omega$ and $K=1$, it gives for $X\ge2$

$$\sum_{n\le X}(\omega(n)-A(X))^2\ll X(A(X)+1), \qquad A(X)=\sum_{p\le X}\frac1p. \tag{1}$$

The condition concerning primes dividing the modulus is vacuous. The function is nonnegative and additive, and $\omega(p^a)=1$. Euler's divergence of the prime reciprocal sum implies $A(X)\to\infty$; the elementary bound $A(X)\le1+\log X$ will also suffice below.

For sufficiently large integers $j$, put

$$X_j=2^{j+2},\quad A_j=A(X_j),\quad H_j=\lceil A_j^{3/4}\rceil, \quad D_j=\{1\le n\le X_j:|\omega(n)-A_j|>H_j\}.$$

By (1) and Chebyshev's inequality,

$$|D_j|\ll X_j\frac{A_j+1}{H_j^2}=o(2^j). \tag{2}$$

For an integer $u\in[2^j,2^{j+1})$, let $b_j(u)$ count those $n\in D_j$ whose image $n+\omega(n)$ belongs to $(u,u+H_j+1)$. Each integer image belongs to at most $H_j+1$ of these windows. Hence

$$\sum_{2^j\le u<2^{j+1}} b_j(u)\le(H_j+1)|D_j|.$$

Set $E_j=\{u\in[2^j,2^{j+1})\cap\mathbb Z:b_j(u)>H_j+1\}$. Then $|E_j|\le|D_j|=o(2^j)$. Both the union $E=\bigcup_jE_j$ and

$$D=\bigcup_j(D_j\cap[2^j,2^{j+1}))$$

have natural density zero. Indeed, for any $\varepsilon>0$ their dyadic counts are at most $\varepsilon2^j$ for all sufficiently large $j$. Summing complete dyadic intervals and bounding the last partial one by its whole interval gives at most $2\varepsilon x+O(1)$ elements below $x$.

If $u\notin E_j$, consider all positive integers $n$ with $n+\omega(n)\in(u,u+H_j+1)$. They satisfy $n\le X_j$ for sufficiently large $j$, since $\omega(n)\ge0$ and $2^{j+1}+H_j+1<2^{j+2}$. The latter inequality follows from $H_j=O((j+1)^{3/4}+1)$. Those $n$ outside $D_j$ obey

$$u-A_j-H_j<n<u+2H_j+1-A_j.$$

There are at most $3H_j+2$ integers in this interval. The exceptional inputs contribute at most $b_j(u)\le H_j+1$. The whole count is thus at most $4H_j+3$.

For real $x\ge1$, put $u=\lfloor x\rfloor$ and choose the dyadic index $j$ with $2^j\le u<2^{j+1}$. Define

$$F(x)= \begin{cases} H_j,&u\notin E_j,\\ 2(u+1),&u\in E_j. \end{cases} \tag{3}$$

For the finitely many initial $u$ not covered by the construction use the second alternative. Both alternatives tend to infinity with $x$. At a nonexceptional location, $(x,x+F(x))\subset(u,u+H_j+1)$, so the normalized count is at most $4+3/H_j\le7$. At an exceptional location, every counted positive integer satisfies $n<x+F(x)<3(u+1)$. Its normalized count is therefore less than $3/2$. This proves the asserted bound at every sufficiently large real location, including nonintegral ones.

Finally, let $G$ be the positive integers outside $E\cup D$, omitting the finitely many initial values. This set has natural density one. For $u\in G\cap[2^j,2^{j+1})$, eventually

$$\omega(u)\ge A_j-H_j\ge A_j/2, \qquad \frac{F(u)}{\omega(u)}\le\frac{2H_j}{A_j}\longrightarrow0.$$

Thus (3) satisfies the density-one width condition but its normalized image count cannot tend to infinity along any sequence of locations. This establishes L34 exactly as stated.