# What do High-Water Marks in Continued Fractions mean?

While reading through several articles concerned with mathematical constants, I kept on finding things like this:

The continued fraction for $\mu$(Soldner's Constant) is given by $\left[1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...\right]$.

The high-water marks are 1, 2, 4, 47, 99, 294, 527, 616, 1152, ... , which occur at positions 1, 2, 3, 12, 70, 126, 202, 585, 1592, ... .

(copied from here)

I didn't find a definition of high-water marks in the web, so I assume that it's a listing of increasing largest integers, while going through the continued fraction expansion.

Is this correct and is there special meaing behind them?

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The convergents at the high-water marks are exceptionally accurate approximations. – Brian M. Scott Jan 21 '12 at 19:39
Probably "high-water mark" means a number larger than any that come before it in the sequence. – Michael Hardy Jan 22 '12 at 2:10

Your definition seems correct to me -- at least, it agrees with the data you've provided and with my intuition, as a native speaker of English, of how this phrase is used. I don't know of a specific significance of these high-water marks, but it's well-known that cutting off a continued fraction expansion just before a particularly large coefficient gives a good rational approximation to the number being expanded. For example the sequence of convergents of π begins 3, 7, 15, 1, 292; the continued fraction [3, 7, 15, 1] is the well-known, surprisingly good approximation 355/113.

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Truncating a continued fraction right before a "high-water" (large) partial quotient $\rm\: a_{i+1}\:$ yields a particularly good rational approximation since

$$\rm r\ =\ [a_0;\ a_1;\ a_2;\ \cdots\ ]\ \ \Rightarrow\ \ \left|\ r\ -\ \frac{p_i}{q_i}\:\right|\ \le\: \frac{1}{a_{i+1}\:q_i^2}$$

For example

$$\rm \pi\ =\ [3;\ 7;\ 15;\ 1;\ 292;\ \cdots\ ]\ \ \Rightarrow \ \ \left|\ \pi - \frac{355}{113}\:\right|\ \le\: \frac{1}{292\cdot 113^2}\: =\ 2.68\cdot 10^{-7}$$

In fact we have $\rm\quad \pi\ -\ \dfrac{355}{113}\ =\ {-}2.67\cdot 10^{-7}\:.$

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The term comes from the common practice of marking bridges or walls with the water level and date during big floods. So, you can look at these high-water marks to see that in 1985, the river flooded to 5 feet above its current level. If you come back a decade later, you may find that a new high-water mark has surpassed it.

I haven't seen the term "high-water marks" for continued fractions before (but I like the name).

I'm sure your interpretation is correct: the $n$th coefficient $a_n$ is a high-water mark if $a_n>a_i$ for $i<n$.

(Maybe it also qualifies if $a_n \geq a_i$ for $i<n$? You would have to find a definition or example to settle that question for sure.)

One significance of a high-water mark at position $n$ is that the $(n-1)$st and $n$th convergents will be big improvements in accuracy. For example, $\pi$ is given by [3;7,15,1,292,...]. In the step before including 292, you go from convergent 333/106=3.14150... to 355/113=3.1415929... When you include 292, you get 103993/33102=3.1415926530..., gaining 2 or 3 more accurate digits each time.

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