# Tag Info

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 ...
Accepted

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 ...

### 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 ...

### 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. (...

### 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 ...

### 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" ...

### 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 ...

### 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 ...
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 ...
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 ...

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