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Results tagged with reference-request
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user 9924
This tag is used if a reference is needed in a paper or textbook on a specific result.
3
votes
Accepted
Looking for a paper of Kemperman on semigroups
For all those interested, a scan of Kemperman's paper can be found here.
- 22.6k
3
votes
Accepted
Differences of consecutive ordered fractional parts
The case $h=0$ is known as the "Three-Distance Theorem"; just google for numerous references or look here for discussion and nice pictures, or here for an interesting historical comment.
A standard r …
- 22.6k
7
votes
Accepted
Additive Combinatorics - reference request
For a somewhat similar argument, see Proposition 2.5 from Alon's 1987 paper ``Subset sums", available here.
- 22.6k
23
votes
Accepted
All polynomials are the sum of three others, each of which has only real roots
To address the original question about the polynomials with complex coefficients.
Given a polynomial $P\in\mathbb C[x]$ of degree $n$, write $P=Q+iR$ with $Q,R\in\mathbb R[x]$, and fix arbitrarily a …
- 22.6k
5
votes
0
answers
86
views
Lines meeting a given set in a unique point
Let $p$ be a fixed prime, and suppose that $S$ is a subset of the affine plane $\mathbb F_p^2$. If $|S|\le p+1$, then by the pigeonhole principle, through any given point $s\in S$ there is a line $L=L …
- 22.6k
17
votes
Accepted
sum, integral of certain functions
For the integral, notice that the expression under the square root is
$$ x(x+1)(x+2)+x(x+2)\sqrt{x(x+2)} = \frac12\,x(x+2)(\sqrt x+\sqrt{x+2})^2. $$
Consequently,
\begin{align*}
\frac1{\sqrt{x(x+ …
- 22.6k
2
votes
Bounding the minimal maximum norm of a solution of a linear system.
I believe you cannot give any general bound, but if the coefficients are integers, this is Siegel's Lemma: a system of $M$ equations in $N$ variables with integer coefficients $b_{ij}$ has an integer …
- 22.6k
1
vote
How to find a set of integers that satisfy certain linear conditions
In a slightly different notation, your question reads as follows:
Given positive integers $m$ and $n$, and a function $r\colon[-n,n]\to{\mathbb Z}_{\ge 0}$ with $r(0)=m$, $r(n)=1$, $r(-n)+\dotsb+r( …
- 22.6k
3
votes
Accepted
A large deviation / binomial coefficients bound
After a little thinking, there is a strikingly simple proof, running as follows.
Dividing through both sides of the inequality by $a^{n-K}$, we see that the larger is $a$, the stronger the estimate i …
- 22.6k
3
votes
2
answers
729
views
A large deviation / binomial coefficients bound
Maple seems to suggest that for any real $a\ge 1$ and positive integer $K$ and $n$ with $K\le n/(a+1)$ one has
$$ a^n + na^{n-1} + \binom{n}{2}a^{n-2} +...+ \binom{n}{K}a^{n-K} \le a^{n-K} e^{nH(K/n …
- 22.6k
3
votes
What is/are the best bound/s on the sum of squares of degrees in a graph?
There is a simple spectral upper bound: namely, denoting by $A$ the adjacency matrix of $G$, and by $\mathbf 1$ the $n$-dimensional all-$1$ vector, we have
$$ \sum_{I=1}^n d_i^2=\|A{\mathbf 1}\|^2 …
- 22.6k
9
votes
Lower bound for the fractional part of $(4/3)^n$
It may be interesting to note that, subject to the ABC conjecture, you have the fantastically good estimate
$$ \left\{ \left( \frac43 \right)^n\right\} \gg_\delta \delta^n,\quad \delta\in(0,1). $$
…
- 22.6k
3
votes
0
answers
89
views
Origins of the ``baby Freiman'' theorem
It is a basic folklore fact from the area of additive combinatorics that a subset $A$ of an abelian group satisfies $|2A|<\frac32\,|A|$ if and only if $A$ is contained in a coset of a (finite) subgrou …
- 22.6k
10
votes
2
answers
459
views
An isoperimetric problem on the hypercube
Fix integer $n\ge 1$, and let $E=\{e_1,...,e_n\}$ denote the standard basis of the vector space ${\mathbb F}_2^n$. Thus, for a set $A\subset{\mathbb F}_2^n$, the sumset $A+E:=\{a+e\colon a\in A,\ e\in …
- 22.6k
13
votes
2
answers
476
views
Roots of lacunary polynomials over a finite field
If $P$ is a polynomial over the field $\mathbb F_q$ of degree at most $q-2$ with $k$ nonzero coefficients, then $P$ has at most $(1-1/k)(q-1)$ distinct nonzero roots.
Does this fact have any stan …
- 22.6k