Asymptotic value of a sequence

Question

Assume a real sequence $1=a_1\leq a_2\le \cdots \leq a_n$, and $a_{i+1}-a_i\leq \sqrt{a_i}$. Does this hold: $$\sum_{i=1}^{n-1} \frac{a_{i+1}-a_i}{a_i} \in O(\log n)$$

Answer 1

Lemma 1: If $1 = a_1 \leq a_2 \leq a_3 \leq \cdots$ and $a_{i+1}-a_i \leq \sqrt{a_i}$ then $$ \sum_{i=1}^{n-1} \frac{a_{i+1}-a_i}{a_i} = \Theta(\log a_n) $$ for all $n$.

Proof: By the assumptions we have

$$ 0 \leq \frac{a_{i+1}-a_i}{a_i} \leq \frac{1}{\sqrt{a_i}} \leq 1, $$

and since

$$ \frac{a_{i+1} - a_i}{a_i} = \frac{a_{i+1}}{a_i} - 1 \tag{1} $$

this is equivalent to

$$ 1 \leq \frac{a_{i+1}}{a_i} \leq 2. \tag{2} $$

If $1 \leq x \leq 2$ then

$$ \log x \leq x-1 \leq \frac{\log x}{\log 2}, $$

so setting $x = a_{i+1}/a_i$ in equations $(1)$ and $(2)$ yields

$$ \log \frac{a_{i+1}}{a_i} \leq \frac{a_{i+1} - a_i}{a_i} \leq \frac{1}{\log 2} \log \frac{a_{i+1}}{a_i}. $$

Summing this over the range $i=1,2,\ldots,n-1$ yields

$$ \log \frac{a_n}{a_1} \leq \sum_{i=1}^{n-1} \frac{a_{i+1}-a_i}{a_i} \leq \frac{1}{\log 2} \log \frac{a_n}{a_1}. $$

$$ \tag*{$\square$} $$

Lemma 2: If $a_i \geq 1$ and $a_{i+1}-a_i \leq \sqrt{a_i}$ then $1 \leq a_i \leq i^2+2$.

Proof: Summing $a_{i+1}-a_i \leq \sqrt{a_i}$ over the range $i=1,2,\ldots,n-1$ yields

$$ a_i - 1 = \sum_{j=1}^{i-1} (a_{j+1} - a_j) \leq \sum_{j=1}^{i-1} \sqrt{a_i} \leq i\sqrt{a_i} $$

and hence

$$ a_i - i\sqrt{a_i} - 1 \leq 0. \tag{3} $$

The parabola $y = x^2 - ix - 1$ lies below the $x$-axis for $1 \leq x \leq \frac{1}{2}\left(i + \sqrt{4 + i^2}\right)$, so equation $(3)$ combined with the assumption $a_i \geq 1$ yields

$$ 1 \leq a_i \leq \left(\tfrac{i}{2} + \tfrac{1}{2}\sqrt{4 + i^2}\right)^2 \leq i^2 + 2, $$

where the last inequality follows from Jensen's inequality.

$$ \tag*{$\square$} $$

Claim: If $1 = a_1 \leq a_2 \leq a_3 \leq \cdots$ and $a_{i+1}-a_i \leq \sqrt{a_i}$ then $$ \sum_{i=1}^{n-1} \frac{a_{i+1}-a_i}{a_i} = O(\log n) $$ for all $n$.

Proof: The result is trivially true for $n=1$ so suppose $n \geq 2$. Combining Lemmas 1 and 2 yields

$$ 0 \leq \sum_{i=1}^{n-1} \frac{a_{i+1}-a_i}{a_i} \leq C\log a_n \leq C\log(n^2+2) \leq D \log n $$

for some constants $C$ and $D$.

$$ \tag*{$\square$} $$

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Intuition

The sum in question behaves in many ways like a "discrete logarithm", and in the sense of Lemma 1 we have something like

$$ \sum_{i=1}^{n-1} \frac{a_{i+1}-a_i}{a_i} \approx \log \frac{a_n}{a_1}. $$

For example, if we double every term of the sequence $(a_n)$ then the values on both sides of the $\approx$ remain unchanged. Further, if $a_n$ is the constant sequence $a_n = a_1$ then both sides of the $\approx$ are equal.

(I'm not sure what the analogue of $\log xy = \log x + \log y$ would be.)

We could try to approach this problem by looking at the smooth analogues of the sequence and sum. The difference $a_{i+1} - a_i$ can be thought of as a discrete derivative and the sum as a discrete integral. So if we can find some function $f$ with $f(n) \approx a_n$ and

$$ f'(n) \approx a_{n+1} - a_n $$

then we might expect that

$$ \sum_{i=1}^{n-1} \frac{a_{i+1}-a_i}{a_i} \approx \int_1^n \frac{f'(x)}{f(x)}\,dx = \log \frac{f(n)}{f(1)} \approx \log \frac{a_n}{a_1}. $$

This observation was what lead me to the approach in this answer.