bio  website  math.ucla.edu/~tao 

location  Los Angeles  
age  39  
visits  member for  5 years, 1 month 
seen  9 hours ago  
stats  profile views  75,743 
Professor of Mathematics at UCLA
2d

comment 
Small residue classes with small reciprocal
The paper linked to by Matt Young appears to have moved to nieuwarchief.nl/serie5/pdf/naw52000014380.pdf (and is entitled "Arithmetic applications of Kloosterman sums", in case it moves again). 
Nov 6 
awarded  Great Answer 
Nov 3 
awarded  Nice Answer 
Nov 3 
comment 
Regularity of random Fourier series
Khintchine's inequality (plus an epsilon of Sobolev embedding) gives $C^{kn/2\varepsilon}$ (where we abbreviate $C^{k,\alpha}$ as $C^{k+\alpha}$) and this is basically optimal. I think this sort of analysis goes back to Paley and Zygmund: journals.cambridge.org/action/… 
Nov 2 
comment 
Simultaneous vanishing of convolutions of Mertens function with itself
I can see that Landau's theorem could be used to deduce $M_k$ changes sign infinitely often, but how could it be used to force it to be zero infinitely often? 
Nov 2 
comment 
Lower bound on number of smooth values of polynomial at primes
For what it's worth: there is at least one error in the proof of Lemma 2 in the arXiv preprint, namely in the third line of (21). An equality of integers is claimed there due to (18), but (18) only gives the claimed equality modulo $p^{i1}(p1)$. This is insufficient to then deduce the next line of (21). 
Nov 2 
comment 
Lower bound on number of smooth values of polynomial at primes
Standard heuristics would suggest that this proportion should be $\rho(d/u)$, where $d$ is the degree of $f$ and $\rho$ is Dickman's function. So, the claimed lemma should fail about $\rho(11) \approx 10^{12}$ of the time, suggesting that counterexamples exist but would require an immense amount of computation to locate numerically. 
Oct 31 
awarded  Good Answer 
Oct 26 
comment 
Do singular values dominate eigenvalues?
A relevant reference here is the classic paper of Horn, ams.org/mathscinetgetitem?mr=61573 , showing that the Weyl inequalities in Yanqi's answer in fact generate the complete set of relations between singular values and eigenvalues. Also, one can prove the inequalities by applying GramSchmidt to the eigenvectors to conjugate $A$ by a unitary matrices to uppertriangular form (so that the eigenvalues become diagonal entries), and then applying the SVD and orthogonality relations for the singular vectors. 
Oct 24 
awarded  Famous Question 
Oct 19 
awarded  Yearling 
Oct 17 
comment 
Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $
If $x \in I_j^n$, then $f(x)  F_n(x) \leq 2^n \int_{I_j^n} f(x)  f(y)\ dy \leq 2^n \int_{t \leq 2^{n}} f(x+t)f(x)\ dt$, by the triangle inequality followed by a change of variable. If one then uses Minkowski's integral inequality, one obtains the desired inequality losing a factor of $2$. It may be possible though to improve the loss of $2$ with a more careful argument. 
Oct 16 
comment 
Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $
One can pointwise upper bound $f(x)F_n(x)$ by a constant times the average of $f(x+t)f(x)$ for $t \leq 2^{n}$, and the upper bound you want then follows from the triangle inequality. 
Oct 16 
comment 
Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $
One may need to make more precise what "best upper bound" means here. Otherwise, the best upper bound for $\fF_n\_{L^p([0,1])}$ will just be $\fF_n\_{L^p([0,1])}$. 
Oct 15 
awarded  Enlightened 
Oct 15 
awarded  Nice Answer 
Oct 15 
revised 
what would be the consequences on the distribution of primes of $\Lambda=\infty$?
added 55 characters in body 
Oct 15 
revised 
what would be the consequences on the distribution of primes of $\Lambda=\infty$?
added 159 characters in body 
Oct 15 
answered  what would be the consequences on the distribution of primes of $\Lambda=\infty$? 
Oct 13 
answered  a road from virial identity to Strichartz estimates for wave/ Schrodinger eqs? 