Thomae's function

Thomae's function, named after Carl Johannes Thomae, has many names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function,[1] the Riemann function, or the Stars over Babylon (John Horton Conway's name).[2] This real-valued function of a real variable can be defined as:[3]

Point plot on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2

Since every rational number has a unique representation with coprime (also termed relatively prime) and , the function is well-defined. Note that is the only number in that is coprime to

It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere.

PropertiesEdit

  • Thomae's function   is bounded and maps all real numbers to the unit interval: 
  •   is periodic with period   for all integers n and all real x.
Proof of periodicity

For all   we also have   and hence  

For all   there exist   and   such that   and   Consider  . If   divides   and  , it divides   and  . Conversely, if   divides   and  , it divides   and  . So  , and  .

  •   is discontinuous at all rational numbers, dense within the real numbers.
Proof of discontinuity at rational numbers

Let   be an arbitrary rational number, with   and   and   coprime.

This establishes  

Let   be any irrational number and define   for all  

These   are all irrational, and so   for all  

This implies   and  

Let  , and given   let   For the corresponding   we have

  and

 

which is exactly the definition of discontinuity of   at  .

  •   is continuous at all irrational numbers, also dense within the real numbers.
Proof of continuity at irrational arguments

Since   is periodic with period   and   it suffices to check all irrational points in   Assume now   and   According to the Archimedean property of the reals, there exists   with   and there exist   such that

for   we have  

The minimal distance of   to its i-th lower and upper bounds equals

 

We define   as the minimum of all the finitely many  

  so that

for all     and  

This is to say, all these rational numbers   are outside the  -neighborhood of  

Now let   with the unique representation   where   are coprime. Then, necessarily,   and therefore,

 

Likewise, for all irrational   and thus, if   then any choice of (sufficiently small)   gives

 

Therefore,   is continuous on  

  •   is nowhere differentiable.
Proof of being nowhere differentiable
  • For rational numbers, this follows from non-continuity.
  • For irrational numbers:
For any sequence of irrational numbers   with   for all   that converges to the irrational point   the sequence   is identically   and so  
According to Hurwitz's theorem, there also exists a sequence of rational numbers   converging to   with   and   coprime and  
Thus for all     and so   is not differentiable at all irrational  
See the proofs for continuity and discontinuity above for the construction of appropriate neighbourhoods, where   has maxima.
  •   is Riemann integrable on any interval and the integral evaluates to   over any set.
The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero.[4] Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval. The function's integral is equal to   over any set because the function is equal to zero almost everywhere.

Related probability distributionsEdit

Empirical probability distributions related to Thomae's function appear in DNA sequencing.[5] The human genome is diploid, having two strands per chromosome. When sequenced, small pieces ("reads") are generated: for each spot on the genome, an integer number of reads overlap with it. Their ratio is a rational number, and typically distributed similarly to Thomae's function.

If pairs of positive integers   are sampled from a distribution   and used to generate ratios  , this gives rise to a distribution   on the rational numbers. If the integers are independent the distribution can be viewed as a convolution over the rational numbers,  . Closed form solutions exist for power-law distributions with a cut-off. If   (where   is the polylogarithm function) then  . In the case of uniform distributions on the set    , which is very similar to Thomae's function. Both their graphs have fractal dimension 3/2.[5]

The ruler functionEdit

For integers, the exponent of the highest power of 2 dividing   gives 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, ... (sequence A007814 in the OEIS). If 1 is added, or if the 0s are removed, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, ... (sequence A001511 in the OEIS). The values resemble tick-marks on a 1/16th graduated ruler, hence the name. These values correspond to the restriction of the Thomae function to the dyadic rationals: those rational numbers whose denominators are powers of 2.

Related functionsEdit

A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible; the set of discontinuities of any function must be an Fσ set. If such a function existed, then the irrationals would be an Fσ set. The irrationals would then be the countable union of closed sets  , but since the irrationals do not contain an interval, nor can any of the  . Therefore, each of the   would be nowhere dense, and the irrationals would be a meager set. It would follow that the real numbers, being a union of the irrationals and the rationals (which is evidently meager), would also be a meager set. This would contradict the Baire category theorem: because the reals form a complete metric space, they form a Baire space, which cannot be meager in itself.

A variant of Thomae's function can be used to show that any Fσ subset of the real numbers can be the set of discontinuities of a function. If   is a countable union of closed sets  , define

 

Then a similar argument as for Thomae's function shows that   has A as its set of discontinuities.

For a general construction on arbitrary metric space, see this article Kim, Sung Soo. "A Characterization of the Set of Points of Continuity of a Real Function." American Mathematical Monthly 106.3 (1999): 258-259.

See alsoEdit

NotesEdit

  1. ^ "…the so-called ruler function, a simple but provocative example that appeared in a work of Johannes Karl Thomae … The graph suggests the vertical markings on a ruler—hence the name." (Dunham 2008, p. 149, chapter 10)
  2. ^ John Conway. "Topic: Provenance of a function". The Math Forum. Archived from the original on 13 June 2018.
  3. ^ Beanland, Roberts & Stevenson 2009, p. 531
  4. ^ Spivak 1965, p. 53, Theorem 3-8
  5. ^ a b Trifonov, Vladimir; Pasqualucci, Laura; Dalla-Favera, Riccardo; Rabadan, Raul (2011). "Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data". Scientific Reports. 1 (191). doi:10.1038/srep00191. PMC 3240948. PMID 22355706.

ReferencesEdit

External linksEdit