Unbounded sequence?

**bkarpuz** · October 4th 2008, 02:50 AM

I have a solution for the following problem but I really wonder what kind of ways will be provided for this.

Problem. Let
$a(m,n)= \begin{cases} 67/256,&\mathrm{mod}(m,4)=0\text{ and }\mathrm{mod}(n,2)=0\\ %\delta 27/256,&\text{ otherwise}. \end{cases}$
Prove that
$ \limsup\limits_{m,n\to\infty}A(m,n)=\infty, $
where
$ A(m,n):=\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n} a(i,j)\Bigg\{\frac{4}{3}\Bigg(\sum\limits_{u=i+1}^ {i+3}a(u,j+1)\Bigg)^{1/4}-1\Bigg\} $
................. $-\frac{1}{3}\Bigg[\sum\limits_{i=m-3}^{m}\sum\limits_{j=1}^{n}\sum\limits_{u=i+1}^{i+ 3}a(u,j+1)+\sum\limits_{i=1}^{m}\sum\limits_{j=n-1}^{n}\sum\limits_{u=i+1}^{i+3}a(u,j+1)\Bigg].$

**Laurent** · October 4th 2008, 04:04 AM

The sequence $a(m,n)$ is "almost" constant. Let's bound $A(m,n)$ from below by replacing $a(m,n)$ by $a=\frac{27}{256}$ in the first term and by $a'=\frac{67}{256}$ in the second one (after the minus sign): we get $A(m,n)\geq mna(\frac{4}{3}(3a)^{1/4}-1)-(12n+6m)a'$ . However, we have $\frac{4}{3}(3a)^{1/4}=\left(\frac{256 a}{27}\right)^{1/4}=1$ so that this lower bound is slightly too strong, but if we had a positive factor instead in front of $mn$ (no matter how small), this would be right. Then, to bound the first term, it suffices to consider the indices where $4|i+1$ and $2|j+1$ (you can bound the terms with other indices by 0 thanks to the previous computation). There are about $\frac{mn}{8}$ such couples $(i,j)$ , and for each of them the summand is greater than $a(\frac{4}{3}(2a+a')^{1/4}-1)$ , which is strictly greater than $a(\frac{4}{3}(3a)^{1/4}-1)=0$ . We get that $A(m,n)\geq \lambda mn- \alpha m - \beta n$ for some positive $\lambda, \alpha,\beta$ . Choosing $m=n$ for instance, you easily see that this is unbounded from above.

**bkarpuz** · October 4th 2008, 04:35 AM

Here is my solution.
Solution. Clearly, we have $a(m+4,n)=a(m,n)$ and $a(m,n+2)=a(m,n)$ for all $m,n\in\mathbb{N}$ .
Therefore, we have
$\sum\limits_{i=m-3}^{m}\sum\limits_{j=n-1}^{n}a(i,j)\Bigg\{\frac{4}{3}\Bigg(\sum\limits_{u =i+1}^{i+3}a(u,j+1)\Bigg)^{1/4}-1\Bigg\}=\frac{81}{256}\bigg(\frac{\sqrt{11}}{3}-1\bigg)>0$ ........(1)
and
$\sum\limits_{i=m-3}^{m}\sum\limits_{j=n-1}^{n}\sum\limits_{u=i+1}^{i+3}a(u,j+1)=3$ .................................................. ...............(2)
for all $m,n\in\mathbb{N}$ .
Therefore, using (1) and (2), we have the following inequalities:
$ \sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}a(i,j)\B igg\{\frac{4}{3}\Bigg(\sum\limits_{u=i+1}^{i+3}a(u ,j+1)\Bigg)^{1/4}-1\Bigg\}\geq\frac{81}{256}\bigg(\frac{\sqrt{11}}{3 }-1\bigg)\bigg\lfloor\frac{m}{4}\bigg\rfloor\bigg\lf loor\frac{n}{2}\bigg\rfloor $ ......(3)
$\sum\limits_{i=m-3}^{m}\sum\limits_{j=1}^{n}\sum\limits_{u=i+1}^{i+ 3}a(u,j+1)\leq3\bigg\lceil\frac{n}{2}\bigg\rceil$ .................................................. .........._(4)
$\sum\limits_{i=1}^{m}\sum\limits_{j=n-1}^{n}\sum\limits_{u=i+1}^{i+3}a(u,j+1)\leq3\bigg\ lceil\frac{m}{4}\bigg\rceil$ .................................................. ..........,(5)
for all $m,n\in\mathbb{N}$ .
Hence, from (3)--(5), we learn that
$A(m,n)\geq B(m,n)$
holds for all $m,n\in\mathbb{N}$ , where
$B(m,n):=\frac{81}{256}\bigg(\frac{\sqrt{11}}{3}-1\bigg)\bigg\lfloor\frac{m}{4}\bigg\rfloor\bigg\lf loor\frac{n}{2}\bigg\rfloor-\Bigg\{\bigg\lceil\frac{n}{2}\bigg\rceil+\bigg\lce il\frac{m}{4}\bigg\rceil\Bigg\}$
for $m,n\in\mathbb{N}$ .
It suffices to prove that $B$ is unbounded.
As $k\to\infty$ , we have
$B(4k,2k)=\frac{81}{256}\bigg(\frac{\sqrt{11}}{3}-1\bigg)k^{2}-2k\to\infty$ .
Thus, the proof is done. $\rule{0.3cm}{0.3cm}$

Notation. $\lfloor\cdot\rfloor$ denotes the least integer function and $\lceil\cdot\rceil$ denotes the greatest integer function.

Appendix. Here, (2) is shown.
By the periodicity, we obtain
$\sum\limits_{i=m-3}^{m}\sum\limits_{j=n-1}^{n}a(i,j+1)=1$ .................................................. .......................(6)
for all $m,n\in\mathbb{N}$ .
Using (6), we get
$\sum\limits_{i=m-3}^{m}\sum\limits_{j=n-1}^{n}\sum\limits_{u=i+1}^{i+3}a(u,j+1)=\sum\limit s_{i=m-3}^{m}\sum\limits_{u=i+1}^{i+3}\sum\limits_{j=n-1}^{n}a(u,j+1)$
......................................_ $=\sum\limits_{u=1}^{3}\sum\limits_{i=m+u-3}^{m+u}\sum\limits_{j=n-1}^{n}a(i,j+1)$
......................................_ $=\sum\limits_{u=1}^{3}1=3$
for all $m,n\in\mathbb{N}$ .

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