*[Thanks — I’ve changed it, and also changed Emmanuel Kowalski’s first comment.]*

A couple of points. Firstly, there is something on lower bounds for this gap problem in the book of Konyagin and Shparlinski, but I forget exactly what they prove.

Secondly, the p^{o(1)} conjecture, even when delta = 1/2, implies the Kakeya conjecture! I wrote some notes on this a long time ago, and here they are:

http://www.dpmms.cam.ac.uk/~bjg23/restriction_kakeya_phenomena/notes10.ps

The connection is basically due to Bourgain from about 1994.

Tim’s “probably inconsistent model” is very close to the idea of doing analytic number theory in which one admits the possibility of a zero of some L-function (or, worse, a Siegel zero). Occasionally one can work through arguments keeping track of the contribution of the hypothetical zero and it can even help (cf. Heath-Brown’s paper proving twin primes on the assumption there is a very strong type of Siegel zero)

Best

Ben

See, e.g., [A. Pagh, R. Pagh, and M. Ruzic. Linear probing with constant independence. In Proc. 39th STOC, pages 318–327, 2007], which shows that hash functions with limited independence achieve good hashing behavior. Here “limited independence” is still rather more independence than you would get by dilating by a single randomly-chosen t, though.

]]>In the opposite direction, but still in this area, Robinson had shown that the general problem would be unsolvable, provided a single particular “exponential” relation could be encoded in polynomial form, and this had apparently been taken to mean that her approach was doomed to fail…(see the famous review MR0133227 (24 #A3061)).

]]>As for the general case, I suspect that, as has been suggested, it may be hard to give a convincing reason for its being impossible to prove the theorem by purely combinatorial means. However, for any given purely combinatorial attack, it may be possible to use the quadratic-residues example to show that it does not work.

I am trying, but so far failing, to think of a problem where it definitely does happen that a single example kills off all hope of solving the general problem by general means. A first question then might be: if you can’t prove it by general means, then how *can* you prove it? A possible answer to that is that you first do some sort of classification and then deal with each class separately. So there might be examples in group theory, say, where it is hopeless to prove a theorem without using CSFG. But that would still leave the question of whether for any given problem there are clear signs that one has to use a classification. There are examples in group theory of classification-free proofs of theorems that had been thought to need classification, so perhaps the answer to this question is no.

In the Tricki article, I wanted to find circumstances where it is *probably* better to concentrate on the special case, so I’d settle for a reasonably convincing and generalizable argument that that is the case here, even if the argument didn’t establish it beyond all possible doubt, and even if one could think of exceptions to the general rule.

It is of course quite difficult, as Timothy points out, to definitively rule out the utility of tackling a general case first before looking at the special case, because there may be ways to exploit the generality that one simply has not thought of yet. The one exception I can see is if the general case has an important “near-counterexample” that forces any proof of that general case to be very delicate, whilst the special case, being somehow “very far” from this counterexample, looks more amenable to cruder and simpler techniques. [Of course, if the general case has a *genuine* counterexample, then it becomes very easy to decide whether to spend any time trying to prove the general case. 🙂 ]

The one exception to my “Never” would be a situation where you can easily deduce the general case from the special case.

]]>It seems to me that any proof of the least quadratic nonresidue problem must, at some point, introduce a new (or at least underused) fact about quadratic residues that was not fully exploited in the past, as it seems that having a large least quadratic nonresidue is perfectly compatible with the “standard” facts we know about residues, though there is no real formalisation of this that I know of. (For instance, my theorem with Ben that the primes contained arbitrarily long arithmetic progressions used extremely little number theory, but it still needed to know something moderately non-trivial about the primes, namely that (roughly speaking) they were a dense subset of a pseudorandom set of almost primes – a fact which had been implicitly used a couple times in the past, e.g. in papers of Ramare and of Green, but perhaps not as systematically as it should have been.)

]]>Actually, that doesn’t mean there couldn’t be a clever combinatorial re-formulation that would lead to a proof of this conjecture. For instance, typically, the standard approaches lead to much more precise results than a bound just for the least quad. non-residue: they lead to equidistribution of residues and non-residues in very small sets; conversely, assuming conjectures on the size of this least quad. non-residue does not imply much about L-functions (as far as I know). So maybe it’s not “hopeless” to go around the L-functions, but at least this special case shows that the problem lies quite deep.

(Maybe an analogy could be drawn with the proof of RH for varieties over finite fields: Grothendieck’s “programme” for that was highly geometric, and maybe it was felt at some point that it was hopeless to look for an arithmetic proof, but in the end Deligne’s work did involve a lot a arithmetic — and a lot of geometry, but it didn’t involve proving first the intermediate results Grothendieck had in mind).

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