Questions tagged [equidistribution]
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67
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Weyl equidistribution for a periodic $L^2$ function
Let $\alpha $ be a fixed irrational number. For a function $g:\Bbb R\to\Bbb C$, define $$g^*(x)=\sup_{N\geq 1} \frac{1}{N} \sum_{n=1}^N |g(x+\alpha n)| ,$$
and assume that there is a constant $C>0$ ...
- 189
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Effective Hecke Equidistribution
In 1918 and 1920 Hecke introduced his L-functions attached to his Grössencharakteren (Hecke characters) and proved they are equidistributed in a sense to made precise momentarily. One can identify ...
- 137
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Equidistribution on subvarieties of $\mathrm{SL}_n(\mathbb{Z})$
$\DeclareMathOperator\SL{SL}$Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $\SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates ...
- 2,143
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Cardinality of the set $\#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$
Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$.
I am interested in upper bound for
$$
M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}
$$
where $N$ ...
- 3,605
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Equidistribution on $\mathrm{SU}_2$
Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant ...
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Reference request - Pillai-Selberg Theorem
I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. ...
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Vanishing exponential sums of fractional parts of polynomials
Let $p$ be an integer polynomial and $k$ be a natural number, both fixed. Is it the case that if
$$C(\alpha) = \sum_{i=1}^k e(\alpha \{p(i)/k\})$$
equals 0, then $\alpha$ is an integer? Here, $e(x) = ...
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Is equal natural density on intervals with matching areas but opposite signs sufficient to use fixed-width part sizes for a simple Riemann sum?
Suppose we have a sequence $\theta_n$ which is dense on $\left(0,2\pi\right)$. Furthermore, if $A=(x,y)\subset(0,\pi)$ and $B=(x+\pi,y+\pi)$ for some $x,y$, and if we define the natural density of a ...
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for which $A\subset \mathbb{N}$ does there exist irrational $\alpha$ for which $\alpha\cdot A \pmod 1$ is not uniformly distributed?
In this question it is discussed that for a set $$A=\{2^n3^m| n,m=0,1,2,\ldots\}$$ there exists an irrational number $\alpha$ such that the set $\alpha\cdot A$ (being naturally enumerated) is not ...
- 104k
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Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed
Let $$(a_n)_{n \in \mathbb{N}} = (1,2,3,4,6,8,9,12,16,18,\cdots)$$ be a sequence that is a listing of the set $$\{2^n3^m \mid m,n \in \mathbb{N}\}$$ We need to find an irrational number $\alpha$ such ...
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Are some numbers more equidistributed than others?
Weyl's theorem says that $n\alpha$ is equidistributed mod 1 for any irrational $\alpha$. One corollary is that, if I consider the fractional part $\{n\alpha\}$ for $n \leq N$, and look at the indices ...
- 1,059
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How do you ensure that the product of two power series is equidistributed?
Let $f(T) = \sum a_n T^n \in \mathbf{F}_p [[ T ]]$ be a power series. We'll say that $f(T)$ is equidistributed if for every $a \in \mathbf{F}_p$, we have
$$\lim_{X \to \infty} \dfrac{1}{X} \cdot \# \{...
- 1,889
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Is the product of two equidistributed power series equidistributed?
Let $f(T) = \sum a_n T^n \in \mathbf{F}_p [[ T ]]$ be a power series. We'll say that the coefficients of $f(T)$ are equidistributed modulo $p$ if for every residue class $a$ modulo $p$, we have
$$ \...
- 1,889
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Continuous variant of the Chinese remainder theorem
Let $x_1,x_2,\ldots, x_k \in [0,1]$ be irrational numbers.
I'm interested in what, if anything, can be said about the values $\{nx_i \bmod 1: n\in \mathbb{N}\}$.
Specifically, I'm interested if there ...
- 359
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Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$
Consider the series
$$
\sum_{n=1}^{\infty} ( \{ n \xi \} - \frac{1}{2})
$$
where $\{ \}$ denotes the fractional part and $\xi$ is some irrational number. It clearly does not converge but can we show ...
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