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The Skolem-Mahler-Lech theorem in algebraic number theory is a significant generalisation of the obvious statement that a polynomial either has finitely many zeroes (in particular, the set of zeroes is bounded), or it vanishes identically. It appeals to me (despite not really being within my areas of expertise) because it is one of the simplest (non-artificial) results I know of which (currently) comes with an ineffective bound – a bound which is provably finite, but which cannot be computed! It appears that to obtain an effective result, one may need a rather different proof. (I thank Kousha Etessami and Tom Lenagan for drawing this problem to my attention.)
Ineffective bounds seem to arise particularly often in number theory. I am aware of at least three ways in which they come in:
- By using methods from soft (infinitary) analysis.
- By using the fact that any finite set in a metric space is bounded (i.e. is contained in a ball of finite radius centred at a designated origin).
- By using the fact that any set of finite diameter in a metric space is bounded.
Regarding #1, there are often ways to make these arguments quantitative and effective, as discussed in my previous post. But #2 and #3 seem to be irreducibly ineffective: if you know that a set A has finite cardinality or finite diameter, you know it has finite distance to the origin, but an upper bound on the cardinality or diameter does not translate to an effective bound on the radius of the ball centred at the origin needed to contain the set. [In the spirit of the preceding post, one can conclude an effective "meta-bound" on such a set, establishing a large annulus in which the set has no presence, but this is not particularly satisfactory.] The problem with the Skolem-Mahler-Lech theorem is that all the known proofs use #2 at some point.