Samose Seminar Series: Irrational Numbers and Fourier Analysis

Samose Seminar Series: Irrational Numbers and Fourier Analysis

Date & Time: Tuesday, April 10th at 4pm (Tea/Coffee at 3.45pm)

Venue: Simons Centre, Ground Floor

Speaker: David Farris, Simons Centre for the Study of Living Machines, NCBS


Consider any irrational number, e.g. the square root of 2=1.414...=a.  Now consider the sequence of numbers a,2a,3a,4a,5a,...=1.414...,2.828...,4.242..., 5.656..., 7.071....  Now consider only the fractional part, i.e. strip off the integer part of each to get a number between 0 and 1, and we obtain the sequence .414..., .828...,.242...,.656..., .071... , which we can call b,2b,3b,4b, etc.

Claim: this sequence of numbers b,2b, 3b, ... is "equidistributed" between 0 and 1, which means that if we consider, say, a subinterval like (.6,.7), then one tenth of the numbers in this sequence will lie in this interval, because the length of (.6,.7) is one tenth of the length of the whole interval (0,1).

This is a straightforward and, I hope, plausible statement.  But see if you can prove it with your bare hands--it's not obvious!  For comparison, if you instead look at the fractional part of *powers* of an irrational number, i.e. c,c^2,c^3,c^4, the analogous statement is false.

In this talk, I will explain how to prove this result using a remarkable argument by Hermann Weyl, which puts the numbers of this sequence in the complex plane by considering the sequence e^ib,e^(2ib),e^(3ib),...  This will allow us to apply Fourier analysis, but all we actually need is the statement that Fourier series exist, so we may view it as a black box! All that is needed to understand the argument is familiarity with complex exponentials.

This lecture is an apertif for an irregular series of lectures I'll be giving at NCBS on and around complex numbers, Fourier analysis and symmetry.  These are rich sources of ideas for many models and problems that, on the surface, have nothing to do with complex numbers, but can often be transformed into an easier to solve problem by using tools that aren't available in the original statement. The goal is to recognize the kinds of situations in which this kind of analysis may be fruitfully deployed.

Warm up exercise: What happens above if you take a to be a rational number instead?  Take a=3/7, say.  And what can you say about e^ia, e^2ia, e^3ia, ...?

All are welcome!

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