32,090 reputation
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bio website math.ucla.edu/~tao
location Los Angeles
age 39
visits member for 5 years, 1 month
seen 9 hours ago
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/naw5-2000-01-4-380.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 $C^{k-n/2-\varepsilon}$ (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 $p^{i-1}(p-1)$. 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/mathscinet-getitem?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 Gram-Schmidt to the eigenvectors to conjugate $A$ by a unitary matrices to upper-triangular 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 $\|f-F_n\|_{L^p([0,1])}$ will just be $\|f-F_n\|_{L^p([0,1])}$.
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?