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bio website math.ucla.edu/~tao
location Los Angeles
age 39
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Professor of Mathematics at UCLA

May
18
awarded  Popular Question
May
16
awarded  Nice Answer
May
15
comment “Typical” convergence rate for the von Neumann mean ergodic theorem
If one wants quantitative forms of the mean ergodic theorem for arbitrary shift operators $U$, then the correct question to ask is not to bound the convergence rate, but rather to bound the metastability of the averages. See e.g. andrew.cmu.edu/user/avigad/Talks/dc.pdf . The prototypical result in which metastability is quantitative but convergence rate is not is the basic real analysis fact that all bounded monotone sequences converge: see terrytao.wordpress.com/2007/05/23/… .
May
15
comment A Question about Palindromic Numbers and System of Arithmetic Progression
One can still produce counterexamples. For instance, one cannot have $t$ and $2001t+1000$ both be palindromes, because then they would have the same first three digits, while having a ratio somewhere between 2001 and 3001, which one can easily check to be absurd. It might be a good exercise for you to experiment with further counterexamples of this type in case you wish to impose further restrictions on p,q,j,k.
May
15
comment Second differences of primes determined by increasing first differences: every positive even integer?
This is basically a more complicated variant of Polignac's conjecture en.wikipedia.org/wiki/Polignac%27s_conjecture . Given that the latter conjecture is already hopelessly difficult, I don't see why one would expect to be able to resolve the current question, although the same probabilistic heuristics that strongly suggest Polignac's conjecture to be true would also apply here to suggest that this conjecture is probably true as well.
May
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awarded  Guru
May
14
answered A Question about Palindromic Numbers and System of Arithmetic Progression
May
14
revised Why do people use “formal calculation” to describe informal calculations?
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May
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revised Why do people use “formal calculation” to describe informal calculations?
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14
awarded  Good Answer
May
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revised Why do people use “formal calculation” to describe informal calculations?
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May
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revised Why do people use “formal calculation” to describe informal calculations?
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awarded  Enlightened
May
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awarded  Nice Answer
May
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answered Why do people use “formal calculation” to describe informal calculations?
May
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comment Random sequence of integers in $\{1, 2, \dots, n \}$ which is “everywhere probably increasing” - how long can it be?
Nice! I had considered using a Fourier expansion to try to diagonalise the $a < b$ constraint, but dropped the idea after seeing the $1 \leq a$ and $b \leq n$ constraints also, though the neat "carrying the 1" decomposition deals with these constraints quite elegantly.
May
11
comment Random sequence of integers in $\{1, 2, \dots, n \}$ which is “everywhere probably increasing” - how long can it be?
In practice, dyadic arguments tend to be non-optimal by a factor of roughly 2 or so, but they have the advantage that the best constants for the dyadic version are often computable explicitly, even if they don't match the best constants for the continuous problem. Unfortunately most of the tools in my bag of tricks aren't geared towards extracting optimal constants for continuous problems...
May
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comment Random sequence of integers in $\{1, 2, \dots, n \}$ which is “everywhere probably increasing” - how long can it be?
It may be that a Bellman function technique (yet another tool from the harmonic analysis toolbox!) may be useful in reducing the constant further, particularly if one is able to phrase the problem in a suitably "dyadic" formulation.
May
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comment Random sequence of integers in $\{1, 2, \dots, n \}$ which is “everywhere probably increasing” - how long can it be?
If one uses the dyadic Hilbert inequality $\sum_{I,J \hbox{ adjacent}} \sum_{i \in I} \sum_{j \in J} \frac{c_i d_j - c_j d_i}{|I|} \leq 2 (\sum_i c_i^2)^{1/2} (\sum_j d_j^2)^{1/2}$ (easiest to prove using the Haar wavelet basis) it appears that the constant $\pi \log 2$ can be reduced slightly to 2.
May
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awarded  Enlightened