Linked Questions
11 questions linked to/from How small can a sum of a few roots of unity be?
23
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5
answers
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Smallest non-zero eigenvalue of a (0,1) matrix
What's the smallest absolute value possible of a non-zero eigenvalue of an $n$ by $n$ square matrix whose entries are either $0$ or $1$ (all operations are over $\mathbb{R}$)? I would be interested ...
- 3,377
11
votes
2
answers
1k
views
Heuristic lower bounds on small sums of roots of unity
Let $f(k,n)$ be the smallest non-zero absolute value of a sum of $k$ complex $n$th roots of unity. Asking for bounds in either direction, Tao suggested that a polynomial lower bound seemed plausible ...
- 4,589
8
votes
5
answers
1k
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Small values of a polynomial evaluated at roots of unity
The MO answer https://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant $C(\...
- 47.4k
4
votes
2
answers
621
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Character values bounded away from zero
Character values for a finite group are sums of nth roots of unity. I'm wondering if there are any results bounding nonzero values of irreducible characters away from zero. Or if not are there ...
4
votes
1
answer
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Are all sums of subsets of roots of unity unique?
For a prime $p$, let $S$ be the set of all $p$-roots of unity in the complex plane. Now, consider the sum, $W(R)$ of the members of a set $R$ which is a proper subset of $S$. I suspect that $R \ne R'$ ...
- 143
4
votes
2
answers
654
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Non-standard Gauss sums
I have the following problem. Let $p$ be some prime. What is the value of
\begin{equation}
\sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl},
\end{equation}
where $\left(\frac{k+1}{p}\right)$ ...
- 145
5
votes
0
answers
370
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Lower bound for some sums of roots of unity
Let $n$ be a positive integer (assume $n$ is prime for simplicity), and let $x_k = \pm1$, for $k = 0,1,2,..., n-1$. Let $\rho$ be an $n-$th primitive root of unity, I am interested in a lower bound ...
- 1,101
4
votes
0
answers
303
views
power series and roots of unity
Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series ...
- 205
4
votes
0
answers
281
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How small can the nonzero sum of $O(\log n)$ distinct $n-$th roots of unity be?
The OEIS sequence oeis.org/A108380 gives the least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude.
This sequence seems to imply that the least number of ...
- 10.4k
4
votes
0
answers
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An Optimization Problem for Exponential Polynomials
Let $\omega$ be a primitive complex $n^{th}$ root of unity. I am interested in the following quantity
$$
\max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1}
\left| 1+\omega^k+\omega^{2k}+\...
- 10.4k
1
vote
0
answers
81
views
Expectation of moduli of roots
For a complex polynomial $\sum_0^n a_i z^i$ the sum of roots is readily given by the Vieta's formula. However, after trying to find out the sum of moduli of roots in terms of the coefficients, I feel ...
- 826