31,675 reputation
7140227
bio website math.ucla.edu/~tao
location Los Angeles
age 39
visits member for 5 years
seen 2 days ago
Professor of Mathematics at UCLA

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?
Oct
10
comment Why do roots of polynomials tend to have absolute value close to 1?
... for the purposes of rigorous argument rather than heuristics, it turns out that the Jensen formula argument sketched in my previous comment is more robust (basically because $\log |z|$ diverges at zero far more slowly than $\frac{1}{z}$) and is easier to make fully rigorous, as is done in the paper linked to in my previous comment.
Oct
10
comment Why do roots of polynomials tend to have absolute value close to 1?
Basically, the randomness is needed to ensure that the denominator $a_n z^n + \dots + a_0$ does not inconveniently end up being unexpectedly small on the contour of integration, which would make the error in the approximation unpleasantly large. (To ensure that this unexpected smallness does not occur requires a bit of work, and is part of what is now known as Littlewood-Offord theory, introduced by Littlewood and Offord to study almost exactly the problem discussed here, namely to understand the distribution of roots of random polynomials.)
Oct
8
comment Reverse Hausdorff Young for nonnegative functions
In fact, it is very rare for the Hausdorff-Young inequality to be close to sharp; the function $f$ basically has to look like the indicator function of the sum of a generalised arithmetic progression and an ellipsoid. A precise formulation of this fact (which comes from additive combinatorics) can be found in Proposition 6.4 of this paper of Christ: arxiv.org/pdf/1406.1210.pdf
Oct
7
comment Reverse Hausdorff Young for nonnegative functions
If the spatial domain is the integers (so the frequency domain is the unit circle), then one can take $f$ to be the indicator function of a lacunary sequence such as $1, 2, \dots, 2^n$ and use Rudin's inequality (see e.g. Lemma 4.33 of my book with Van Vu) to contradict the inequality; the case $p = 4/3$ can be worked out by hand for instance. One can then transfer to Euclidean spaces by standard methods (e.g. blurring each integer by an approximation to the identity to pass from ${\bf Z}$ to ${\bf R}$).
Oct
4
comment Sieving question
In particular, taking the product of a prime $p$ between $X^{1/2-\varepsilon}$ and $X^{1/2}$ together with a quadratic residue modulo that prime of size at most $X/p$ already gives a positive density of integers with the required property. (One can also take some primes above $X^{1/2}$, but one starts needing information about the distribution of quadratic residues in short intervals, e.g. Polya-Vinogradov or Burgess bounds.)
Oct
3
comment Why do roots of polynomials tend to have absolute value close to 1?
A slight variant of this argument: Jensen's formula tells us that the zeroes of $p$ occur when $\log |p|$ fails to be harmonic. But $\log |p(z)|$ is typically $O(1)$ for $|z| \leq 1-\varepsilon$ and typically $n \log |z| + O(1)$ for $|z| \geq 1+\varepsilon$, so the main opportunity for non-harmonicity is near the unit circle. This formulation of the argument has the advantage of extending to other models of random polynomials than Kac polynomials, e.g. Weyl polynomials. It can also be pushed to give local universality: see my paper with Van at arxiv.org/abs/1307.4357
Oct
2
comment Has philosophy ever clarified mathematics?
Often the clarifying role of philosophy has been in a double negative sense: good philosophy can help mathematics by preventing bad philosophy from muddying the conceptual waters (e.g., to pick just one such confusion that is no longer an issue, the philosophical controversy about the ontological status of non-Euclidean geometry).
Sep
30
awarded  Explainer
Sep
25
awarded  Favorite Question
Sep
20
awarded  Favorite Question