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Jun
16 |
comment |
Functions about integers which are divisible by all numbers less than or equal to a fractional power of the integer
There's an obvious (and unique) answer to your question : $f(n)$ is the smallest $m$ such that all integers $\leq \sqrt[m]{n}$ divide $n$. |
Jan
28 |
comment |
Counting square-free numbers smoothly
There must be some typo since the sum in the question is $+ \infty$. |
Jan
11 |
awarded | Necromancer |
Jan
11 |
revised |
Elementary applications of Krein-Milman
deleted 40 characters in body |
Jan
10 |
awarded | Teacher |
Jan
10 |
answered | Elementary applications of Krein-Milman |
Dec
20 |
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Existence of non-trivial affine functions on Hadamard spaces ?
Yes, the "good" question was indeed "which Hadamard spaces have this property?". The comment of Igor Belegradek gives exactly what I was looking for. Thanks all. |
Dec
19 |
asked | Existence of non-trivial affine functions on Hadamard spaces ? |
Oct
28 |
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Test functions with small support and nonnegative Fourier transform
An unexpectedly simple solution ; thanks ! I think this still applies in the noncommutative case (the Haar measure is left and right invariant) : If $VV \subset U$, $g_1$ is any map vanishing outside $V$, with nonzero mean, and $g_2 : x \mapsto g_1^{*}(x^{-1})$, then $f = g_1 \star g_2$ should work. |
Oct
28 |
asked | Test functions with small support and nonnegative Fourier transform |
Oct
24 |
accepted | The identity element of a compact group is a limit point of any “polynomial sequence” |
Oct
24 |
comment |
The identity element of a compact group is a limit point of any “polynomial sequence”
I accept this answer since this is close to what I was looking for. It seems reasonable to expect that your proof extends to the general case (without applying the reduction given by Terence Tao) : given a neighbourhood $U$ of the identity, all what we need is a function $f$ with real, nonnegative fourier transform, with nonzero mean, and which vanishes outside $U$. I don't know if this is always possible. |
Oct
23 |
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The identity element of a compact group is a limit point of any “polynomial sequence”
Thanks. There's an argument (also due to Bergelson I think !) which shows that Van der Waerden's theorem implies the theorem in my question : if $U$ is a small neighbourhood of the identity, $G$ has a finite covering by some $(g_iU)_i$, and any map $n\in \mathbb{N} \mapsto$ some $i$ with $g^{P(n)} \in g_iU$ defines a colouring of the integers to which Van der Waerden's theorem can be applied. On then concludes using equations like $(x+2y)^2-2(x+y)^2+x^2=2y^2$ and its generalizations. But since I've never considered Van der Waerden's theorem as "elementary" ... |
Oct
23 |
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The identity element of a compact group is a limit point of any “polynomial sequence”
@GH: when $G$ is a torus, one can conclude by a standard Weyl's sums argument. But by "a straightforward generalisation" I meant "replace the torus by $G$ throughout the proof" (since noncommutativity causes no trouble in our ultralimits). @Terry Tao: Thanks for the elegant reduction. Your argument using Peter-Weyl theorem works also in the noncommutative case (which could arise if one considers instead sequences $n \mapsto g^{P(n)}h^{Q(n)}$ - where the ultralimit argument still applies), in which case we only have to consider (a finite list of) compact Lie groups. |
Oct
23 |
comment |
The identity element of a compact group is a limit point of any “polynomial sequence”
For example : this is a straightforward generalisation of theorem 7.2 in the following article of Vitaly Bergelson math.osu.edu/~bergelson.1/VBContempMathUltrafiltersEtc.pdf . |
Oct
23 |
asked | The identity element of a compact group is a limit point of any “polynomial sequence” |
Jul
20 |
comment |
A mixing property of linear map over finite fields
Thanks ! It's amusing that the same constant $\frac{q^2}{2q-1}$ appears here and also as the limit of the polynomial method in the Kakeya problem. |
Jul
20 |
awarded | Scholar |
Jul
20 |
accepted | A mixing property of linear map over finite fields |
Jul
20 |
awarded | Editor |