67
votes
Accepted
For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements
Here is a counterexample. We first need a "more sums than differences" construction:
Lemma. For any $\varepsilon>0$ there exists a cyclic group ${\bf Z}/N{\bf Z}$ and a non-empty subset ...
- 98.5k
17
votes
Accepted
Integer solutions of an exponential equation
A general method, not necessarily best for this particular equation, is to split into three cases by writing $x=3u+v$ with $v\in\{0,1,2\}$. Then rewrite your equation as
$$ A(7^u)^3 + 2 = y^2\quad\...
- 44.2k
10
votes
Kissing number lower bound vs. upper bound - precise meanings?
Lots of questions here, I'll see how many I can address. To be clear, $K_L$ and $K_U$ are summaries of our current knowledge. There is some true kissing number in each dimension, and hopefully we'll ...
- 145k
8
votes
Integer solutions of an exponential equation
Here's a proof that the $x=1$ solution is unique
using only facts about "Pell equations" that were already known to Fermat
(if not centuries earlier to Bhaskara II et al.)
and should ...
- 75.2k
7
votes
Accepted
Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?
The correct viewpoint is not "$\Lambda$ is like a disc", but "$\Lambda$ is like the functions on a disc".
To see this, ask yourself: given an element $f \in \mathbb{Z}_p[[T]]$, ...
- 34.4k
6
votes
Accepted
Faithful representations of integral models
Yes, there exists a closed immersion $\mathcal{G}\to \mathrm{GL}_n$ over
$\mathbb{Z}$. This is folklore. For a proof, see for example Proposition 3 of arXiv:2012.05708v3
- 96
6
votes
Accepted
Is the value of the power series at 0.1 transcendental?
This question is likely open.
We can tell whether $f(1/10)$ is rational or irrational (by asking whether $a_n$ is eventually periodic); in this case, definitely irrational.
Can we have an algebraic ...
- 39.4k
4
votes
Sum of $\frac{1}{(\delta_1,\delta_2)}$ with congruence restrictions
Your sum is
\begin{align*}
L(Q)&=\sum _{d\leq Q}\frac {1}{d}\sum _{\delta _1,\delta _2\leq Q/d\atop {a|d\delta _1\atop {b|d\delta _2\atop {(\delta _1,\delta _2)=1}}}}1=\sum _{dh\leq Q}\frac {\mu (...
- 1,038
3
votes
Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?
I give this answer just to make David's answer a little more general. Precisely, your $\Lambda$ should be thought as functions with norm less than $1$ in some Banach $\mathbb{Q}_p$-algebras. We fix a ...
- 436
3
votes
Accepted
Lang's remark on Lindemann-Weierstrass theorem
For brevity of notation, if $I = (i_1,\ldots,i_n) \in \mathbf N^n$, write $x^I = x_1^{i_1}\cdots x_n^{i_n}$. Write $\boldsymbol \alpha$ for $(\alpha_1,\ldots,\alpha_n)$, and set $I \cdot \boldsymbol \...
- 24.4k
3
votes
Asymptotic density of an infinite union of subgroups
It's false. Up to an (irrelevant) factor of $2$, I work with $\mathbb{N}$ instead of $\mathbb{Z}$.
The only ingredient needed is that $c_D \to 0$ as $D \to \infty$, where $c_D > 0$ is such that $$\...
- 643
3
votes
Accepted
Special configurations on a circle from a homological algebra problem
There is a simple characterization of interesting configurations:
Lemma. A configuration $x_0=0< x_1 < x_2 < ... <x_r$ of Gorenstein dimension $g$ is interesting if and only if there exist ...
- 28.7k
3
votes
On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?
Stupid of me. As O. Gorodetsky mentions, these are classical:
$$F_1=(91\zeta(3)-2\pi^3\sqrt{3})/432$$
$$F_2=(28\zeta(3)-\pi^3)/64$$
$$F_3=(117\zeta(3)-2\pi^3\sqrt{3})/243$$
In addition, note that ...
- 10.4k
3
votes
The theta function of an odd Dirichlet character
The point is that $\theta_\chi(t)$ is a modular form whose coefficients are essentially $\chi(n)$. If we twist $\chi(n)$ by some random odd smooth function of $n$, we don't get a modular form. (...
- 90.6k
3
votes
Is the value of the power series at 0.1 transcendental?
For the UPDATE, allowing coefficients $a_n < M$ for a fixed $M$. Then there are examples with $f(1/10)$ rational.
Let's do this. Define a sequence $(a_n)$ of coefficients as follows: Start with ...
- 39.4k
2
votes
Exponential sum with weight in bottom
Under your assumption that no $n$ from $1$ to $X$ has $|1 -e(c_1n)|<\epsilon$, we have an upper bound for your sum of the form $2 \epsilon^{-1} \log X + O(X)$.
This is the "trivial bound" ...
- 126k
2
votes
Leech lattice shortest vector vs other 23 cases and E8 cases
Leech lattice $Λ24$ has larger distance between centers of the two balls (namely $D=2$), in contrast with other 23 classes of 24-dimensional lattice which as $D=√2$, also in contrast with the E8 ...
- 126k
2
votes
Number of solutions of $am \equiv bn \pmod{q}$
I made a fatal mistake in my attempt at obtaining a simple asymptotic formula for the number $J$ of solutions to the congruence with the given restrictions. Undaunted, I come back with the proof of a ...
- 8,462
1
vote
Accepted
Iterated exponential sums
This sum is equal to
$$
\frac{\left( \sum_{n \leq x} e(f(n)) \right)^2 - \sum_{n \leq x} e(2 f(n))}{2}
$$
In most situations, we do not expect there to be more than square root cancellation, and thus ...
- 2,314
1
vote
On Zagier's missing continued fraction with multiple limits?
To complete the 12 cfracs in this post and the 4 in the next, all associated with 16 "sporadic sequences", then 13 of them have closed-forms, 1 has six limits (also with closed-forms but one ...
- 11.3k
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