# The Lumber Room

"Consign them to dust and damp by way of preserving them"

## Are there Fibonacci numbers starting with 2012? (continued)

Almost 8 months ago I left the first part of this post unfinished planning to complete it in the morning; seems I slept too long. (Or as this guy said after a 2-year hiatus: “Sorry for the pause guys. I was in the bathroom.”)

Recall: To get a power of 2 that starts with a prefix $p$ (like $p = 2012$), we want $n$ such that the fractional part of $n\log 2$ lies between those of $\log p$ and $\log(p+1)$ (all logarithms here to base 10), and similarly to get a Fibonacci number starting with $p$, we want (with some hand-waving) $n$ such that the fractional part of $n\log\phi$ lies between those of $\log(p) + \log{\sqrt{5}}$ and $\log(p+1) + \log{\sqrt{5}}$. The more general problem is:

Problem: Given an irrational number $\theta$ and an interval $(a,b) \subset (0,1)$, find $n$ such that $\mathrm{frac}(n\theta)$ lies in the interval $(a,b)$.

Here is one method, based on Edward Burger’s book Exploring the Number Jungle. Let $\alpha$ be the midpoint of the interval $(a,b)$. Then we are trying to find $n$ such that $\mathrm{frac}(n\theta)$ is close to $\alpha$.

• Find a fraction $\displaystyle \frac{p}{q}$ approximating $\theta$, such that $\displaystyle |q\theta - p| < \frac1q$. (These are the convergents of the continued fraction of $\theta$, but in practice it seems you can also get away with taking semi-convergents that may not satisfy this property.)
• Let $N$ be the closest integer to $q\alpha$. Note that this automatically means $\displaystyle |q\alpha - N| \le \frac12$
• Write $\displaystyle N = yp - xq$ with $\displaystyle |y| \le \frac{q}{2}$. This you can do quite easily with the Euclidean algorithm.
• Then for $n = q + y$ and $k = p + x$, we have (it is a simple exercise to prove this)
$\displaystyle |\theta n - k - \alpha| < \frac{3}{n}$
• This means that the distance between $n\theta$ and $\alpha$ is small, modulo 1. If this distance turns out to be still too large, start with a bigger convergent $\frac{p}{q}$.

I think I had some code to post as well (hey, what’s the actual Fibonacci number that starts with 2012?), but it needs to be cleaned up… probably this works (in Sage).

The thing we’re doing here is called inhomogeneous Diophantine approximation.

[Originally posted on math.stackexchange, here and here.]

Written by S

Fri, 2012-08-03 at 01:56:18 +05:30

Posted in mathematics

## ABBA’s The Day Before You Came

[A bit too much on a stupid pop song. Move along. :-)]

The Day Before You Came is the last song that ABBA recorded. It is interesting for more than being their swansong: it is also highly atypical of ABBA.

Here is the song (Link is to Dailymotion because Youtube has videos only of live perfomances):

The music and the video are non-ABBAish too (gone are the exuberance and the outlandish clothes, the video is almost entirely Agnetha with the others getting only a few seconds of screen time and no action), but confining ourselves to the lyrics:

Must have left my house at eight, because I always do
My train, I’m certain, left the station just when it was due
I must have read the morning paper going into town
And having gotten through the editorial, no doubt I must have frowned
I must have made my desk around a quarter after nine
With letters to be read, and heaps of papers waiting to be signed
I must have gone to lunch at half past twelve or so
The usual place, the usual bunch
And still on top of this I’m pretty sure it must have rained
The day before you came

I must have lit my seventh cigarette at half past two
And at the time I never even noticed I was blue
I must have kept on dragging through the business of the day
Without really knowing anything, I hid a part of me away
At five I must have left, there’s no exception to the rule
A matter of routine, I’ve done it ever since I finished school
The train back home again
Undoubtedly I must have read the evening paper then
Oh yes, I’m sure my life was well within its usual frame
The day before you came

Must have opened my front door at eight o’clock or so
And stopped along the way to buy some Chinese food to go
I’m sure I had my dinner watching something on TV
There’s not, I think, a single episode of Dallas that I didn’t see
I must have gone to bed around a quarter after ten
I need a lot of sleep, and so I like to be in bed by then
I must have read a while
The latest one by Marilyn French or something in that style
It’s funny, but I had no sense of living without aim
The day before you came

And turning out the light
I must have yawned and cuddled up for yet another night
And rattling on the roof I must have heard the sound of rain
The day before you came

It is putatively a love song, but it makes no explicit declaration of love. The entire lyrics of the song are merely a catalogue of an average day’s events. The song is a timetable. You’ve got to admire the sheer cheek of this, if nothing else. :-) (a la Peter Cushing lives in Whitstable.)

Yet it is interesting. I think this could be argued to be a first-class example of svabhāvokti, the achievement of a poetic effect by simply and ably describing things as they are. The author of the song describes her usual boring routine, presumably to contrast against her much-changed life after meeting the “you” of the song. This is an inversion of the much more common poetic convention, that of recalling time spent in love, time spent together, etc. (Bilhana’s चौरपंचाशिका Chaura-panchashika comes to mind.)

It is linguistically interesting as well: with few exceptions, the verbs are all accompanied by “must have”, “I’m certain”, “undoubtedly”, “I’m sure”, or “I think”. [I'm not sure what this characteristic of the verb is called. In the old days I think we'd just call it verb tense, but now "tense" is reserved for verb forms that indicate time, and "tense/aspect/mood" is used instead. I don't know what this particular "must have"-type of construction is called: a Google search throws up terms like near-certainty mode, deductive, non-factual, evidential, presumptive, etc.] It’s as if the author isn’t sure. (It is remarkable that, in general, adding “I’m certain” or “undoubtedly” makes a statement less certain.) One interpretation is that the narrator after meeting her lover no longer remembers her life from before; so different it has become. But considering the level of detail, is this really possible? Besides, how much can a daily routine really change? The trains will still run at the same time, at any rate. A different possibility suggests itself: that the narrator is simply unreliable.

Suddenly a lot of things make sense: she is not describing her life “before you came”. She hasn’t met anyone at all, but is instead hoping to meet someone that will turn her colourless life exciting. The song is not reminiscences about a dull past (who would want to do that?), but she is instead imagining how different she will feel in the future after finding someone, yet so sad is her case that all she can imagine and describe is her present. The video lends credence to this idea: it starts with a scene of her daily commute, a guy appears and the video goes into a (presumably) imagined mode, before the guy disappears and she’s back at the same train station.

This would finally explain why the prevailing mood of the song, as experienced by the listener, is not one of love (it does not evoke śṛṅgāra शृङ्गार in other words): instead it is one of melancholy and weariness.