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It is easy to produce rank of 2n/3 and probably less than log(n) for n sufficiently large. Take a "diagonal" subspace of R^n that includes the all ones vector, for example. Gerhard "Many Ways To Lose Weight" Paseman, 2015.06.22
Terry Tao has part of his blog devoted to writing. He includes links to many other opinions, including notes on a course by Knuth and others on mathematical writing. If the original poster has exhausted that resource and still has a question on what to do, they might then come back here with a more specific request than "Any ideas?" Gerhard "The Internet Has Your Answer" Paseman, 2015.06.16
Can one quickly compute $\binom{2p}{p} \bmod (2p+1) $ using previous computed smaller values? Do I need to loop from 1 to p, or can I use the factorization of 2p+1 to speed up the computation? Gerhard "Often Looking For Faster Ways" Paseman, 2015.06.12
Further, it is not clear that there isn't some p with 2p+1 composite in which 2p+1 divides the relation. It may follow from Wilson's Theorem, but I have not made the connection yet. Gerhard "Needs More Coffee, Of Course" Paseman, 2015.06.12
Obligatory reference, to keep others from making the similar mistake: For bivariate polynomials, we have from Friedlander and Iwaniec the polynomial x^2 + y^4 assumes prime values at infinitely many integer pairs (x,y). I know of no other examples that are as nice. Gerhard "Someone Would Have Done It" Paseman, 2015.06.06
I was wondering about your comment. Do you mean "generated by a single finite semigroup"? Otherwise one could take a certain (infinite) relatively free algebra as the generating algebra. Gerhard "Looking For Some Nontrivial Answers" Paseman, 2015.06.06
It's probably going to be similar to the behaviour if you look at an arbitrary homogeneous polynomial in n variables of degree n with all positive coefficients, with possibly a scale factor of (det A)^n involved. Gerhard "Try It With Two Variables" Paseman, 2015.06.04
Point taken. Perhaps one should replace "I" with "many" in the above suggestion. Also, most sales pitches I know don't use the word "Sadly" . Gerhard "Your Closing Pitch Needs Work" Paseman, 2015.05.31
It might be better to personalize the state of affairs, as in "Sadly, the dissertation is in a language I don't know (French)." Gerhard "Appearance Matters (To The French)" Paseman, 2015.05.29
It is related, but not exactly the same. In Joseph's version, k (number of distinct distances) is fixed and n is wanted, whereas the literature you mention seems to me to have n fixed and estimates k given n and d. However, that entry a good place to start. Also, the book titled something like "Unsolved problems in geometry" might have a discrete portion that gets closer to Joseph's question. Gerhard "Looking From The Other End" Paseman, 2015.05.23
It seems he uses most of the devices David Speyer lists in the posts, plus some other observations about representing $2^{67} -1$ as $((u+v)/2)^2 - ((u-v)/2)^2$. Anyway, he did do some sifting and a lot of modular arithmetic. Gerhard "Maybe It Was A Casio" Paseman, 2015.05.22
