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27 December, 2008 in math.CA, math.CO, tricks | Tags: additive combinatorics, calibration, Cauchy-Schwarz, Fourier transform, NLS, scale invariance, Sobolev embedding, test cases | by Terence Tao | 7 comments
Title: Use basic examples to calibrate exponents
Motivation: In the more quantitative areas of mathematics, such as analysis and combinatorics, one has to frequently keep track of a large number of exponents in one’s identities, inequalities, and estimates. For instance, if one is studying a set of N elements, then many expressions that one is faced with will often involve some power of N; if one is instead studying a function f on a measure space X, then perhaps it is an norm which will appear instead. The exponent involved will typically evolve slowly over the course of the argument, as various algebraic or analytic manipulations are applied. In some cases, the exact value of this exponent is immaterial, but at other times it is crucial to have the correct value of at hand. One can (and should) of course carefully go through one’s arguments line by line to work out the exponents correctly, but it is all too easy to make a sign error or other mis-step at one of the lines, causing all the exponents on subsequent lines to be incorrect. However, one can guard against this (and avoid some tedious line-by-line exponent checking) by continually calibrating these exponents at key junctures of the arguments by using basic examples of the object of study (sets, functions, graphs, etc.) as test cases. This is a simple trick, but it lets one avoid many unforced errors with exponents, and also lets one compute more rapidly.
Quick description: When trying to quickly work out what an exponent p in an estimate, identity, or inequality should be without deriving that statement line-by-line, test that statement with a simple example which has non-trivial behaviour with respect to that exponent p, but trivial behaviour with respect to as many other components of that statement as one is able to manage. The “non-trivial” behaviour should be parametrised by some very large or very small parameter. By matching the dependence on this parameter on both sides of the estimate, identity, or inequality, one should recover p (or at least a good prediction as to what p should be).
General discussion: The test examples should be as basic as possible; ideally they should have trivial behaviour in all aspects except for one feature that relates to the exponent p that one is trying to calibrate, thus being only “barely” non-trivial. When the object of study is a function, then (appropriately rescaled, or otherwise modified) bump functions are very typical test objects, as are Dirac masses, constant functions, Gaussians, or other functions that are simple and easy to compute with. In additive combinatorics, when the object of study is a subset of a group, then subgroups, arithmetic progressions, or random sets are typical test objects. In graph theory, typical examples of test objects include complete graphs, complete bipartite graphs, and random graphs. And so forth.
This trick is closely related to that of using dimensional analysis to recover exponents; indeed, one can view dimensional analysis as the special case of exponent calibration when using test objects which are non-trivial in one dimensional aspect (e.g. they exist at a single very large or very small length scale) but are otherwise of a trivial or “featureless” nature. But the calibration trick is more general, as it can involve parameters (such as probabilities, angles, or eccentricities) which are not commonly associated with the physical concept of a dimension. And personally, I find example-based calibration to be a much more satisfying (and convincing) explanation of an exponent than a calibration arising from formal dimensional analysis.
When one is trying to calibrate an inequality or estimate, one should try to pick a basic example which one expects to saturate that inequality or estimate, i.e. an example for which the inequality is close to being an equality. Otherwise, one would only expect to obtain some partial information on the desired exponent p (e.g. a lower bound or an upper bound only). Knowing the examples that saturate an estimate that one is trying to prove is also useful for several other reasons – for instance, it strongly suggests that any technique which is not efficient when applied to the saturating example, is unlikely to be strong enough to prove the estimate in general, thus eliminating fruitless approaches to a problem and (hopefully) refocusing one’s attention on those strategies which actually have a chance of working.
Calibration is best used for the type of quick-and-dirty calculations one uses when trying to rapidly map out an argument that one has roughly worked out already, but without precise details; in particular, I find it particularly useful when writing up a rapid prototype. When the time comes to write out the paper in full detail, then of course one should instead carefully work things out line by line, but if all goes well, the exponents obtained in that process should match up with the preliminary guesses for those exponents obtained by calibration, which adds confidence that there are no exponent errors have been committed.
Prerequisites: Undergraduate analysis and combinatorics.