---

This proves L229 at author-recorded standing. Logarithms are natural. We use the following form of the definition of Hausdorff dimension: if a set admits interval covers with maximum length tending to zero and the sum of the $s$th powers of the lengths tending to zero, its dimension is at most $s$. Dimension is stable under countable unions, in the sense that the dimension of the union is the supremum of the dimensions.

Growth and the sparse integer vector

Let $(a_n)$ be one of the sequences in the statement, and put $D=\{n:a_n\ne p_n\}$. For each fixed $0<\eta<1$, density zero provides, for all sufficiently large $n$, indices outside $D$ in both $[(1-\eta)n,n]$ and $[n,(1+\eta)n]$. Monotonicity and the prime number theorem sandwich $a_n/p_n$ between $1-\eta+o(1)$ and $1+\eta+o(1)$. Thus

$$a_n\sim p_n\sim n\log n. \tag{1}$$

In particular the series converges. For some integer $B\ge1$,

$$a_n\le Bn\log(n+1),\qquad p_n\le Bn\log(n+1) \quad(n\ge1). \tag{2}$$

We first restrict to the class with a fixed $B$; the full class is its countable union over $B$.

For $n\ge2$ put

$$h_n=\frac{p_{n+1}-p_n}{2},\qquad h'_n=\frac{a_{n+1}-a_n}{2},\qquad s_n=\frac{h'_n-h_n}{q_n}. \tag{3}$$

These are integers: $a_n$ is odd for $n\ge2$, and the two position differences $a_n-p_n$ and $a_{n+1}-p_{n+1}$ are divisible by $2q_n$, because $q_n\mid q_{n+1}$. Both $h_n$ and $h'_n$ are positive. The support $E=\{n\ge2:s_n\ne0\}$ is contained in $D\cup\{n:n+1\in D\}$, so $E$ has density zero. Moreover, for $N\ge3$,

$$\begin{aligned} \sum_{n=2}^N q_n|s_n| &\le\sum_{n=2}^N(h'_n+h_n)\\ &=\frac{a_{N+1}-a_2+p_{N+1}-p_2}{2} \le3BN\log N. \end{aligned} \tag{4}$$

For sufficiently large $N$, all $n\ge\lceil\sqrt N\rceil$ satisfy $q_n\ge c\log n\ge(c/2)\log N$. Consequently

$$\sum_{n=\lceil\sqrt N\rceil}^N|s_n|\le C_BN, \qquad C_B=\max(1,6B/c). \tag{5}$$

For the earlier coordinates, (4) and $q_n\ge1$ give the crude but sufficient individual bound $|s_n|\le3BN\log N$.

Counting prefixes uniformly

Fix $0<\rho<1/4$ and an integer $N_0$. Restrict the class further by requiring

$$|E\cap[2,N]|\le\rho N\qquad(N\ge N_0). \tag{6}$$

Every sequence in the class with fixed $B$ belongs to one of these classes for some $N_0$, for each fixed $\rho$.

There are only finitely many choices of $a_1,a_2$ under (2). The first $\lceil\sqrt N\rceil$ coordinates of $(s_n)$ have at most

$$(6BN\log N+1)^{\lceil\sqrt N\rceil}=\exp(o_B(N)) \tag{7}$$

possibilities. For the remaining coordinates, put $M=\lceil C_BN\rceil$. A vector with support of size $t$ has at most $\binom Nt$ choices of support and $2^t$ choices of signs. The number of positive integer $t$-tuples with sum at most $M$ is $\binom Mt$, including the value $1$ when $t=0$. By (5)--(6), an upper bound for their total number is therefore

$$\sum_{0\le t\le\lfloor\rho N\rfloor} \binom Nt\,2^t\binom Mt. \tag{8}$$

For $1\le t\le\rho N$ the usual bound $\binom ut\le(eu/t)^t$ and $M\le(C_B+1)N$ give

$$\binom Nt\,2^t\binom Mt \le\left(2e^2(C_B+1)(N/t)^2\right)^t \le\exp\bigl(Nf_B(\rho)\bigr), \tag{9}$$

where

$$f_B(\rho)=\rho\log\frac{2e^2(C_B+1)}{\rho^2} \longrightarrow0\quad(\rho\downarrow0).$$

The last inequality follows because $u\log(2e^2(C_B+1)/u^2)$ is increasing for $0<u\le1/4$. Equations (7)--(9), with the at most $N+1$ summands in (8), show that the number of possible prefixes through index $N$ is at most

$$\exp\bigl(Nf_B(\rho)+o_B(N)\bigr). \tag{10}$$

Here we may count vectors through gap index $N$, since this only overcounts position prefixes through $N$. Each vector and $a_1,a_2$ determine the corresponding position prefix by (3). Ignoring positivity or congruence restrictions in the upper count causes no difficulty.

Covers of the series values

By (2), the tail after index $N$ is bounded uniformly in this class by

$$0\le\sum_{n>N}\frac{a_n}{2^n} \le C'_B N\log N\,2^{-N} \tag{11}$$

for a constant $C'_B$. Each prefix counted in (10) therefore specifies one interval of this length covering all its series values. For any $s>f_B(\rho)/\log2$, the sum of the $s$th powers of these lengths is at most

$$\exp\bigl(N(f_B(\rho)-s\log2)+o_{B,s}(N)\bigr) \longrightarrow0. \tag{12}$$

Thus the set of series values in the class (2), (6) has Hausdorff dimension at most $f_B(\rho)/\log2$. Taking its countable union over $N_0$ gives the same bound for the entire class with fixed $B$. As $\rho$ can tend to zero, that class has dimension zero. Finally take the countable union over $B$. This proves L229. $\square$