Growth of the "cube of square root" function

Question

Hello all, this question is a variant (and probably a more difficult one) of a (promptly answered ) question that I asked here, at Is it true that all the "irrational power" functions are almost polynomial ?.

For $n\geq 1$, let $f(n)$ denote the "integer part" (largest integer below) $n^{\frac{3}{2}}$, and let $g(n)=f(n+2)-2f(n+1)+f(n)$. Question : Is it true that $g(n)$ is always in $\lbrace -1,0,1\rbrace$ (excepting the initial value $g(1)=2$) ? I checked this up to $n=100000$.

It is not too hard to check that if $t$ is large enough compared to $r$ (say $t\geq \frac{3r^2+1}{4}$), $f(t^2+r)$ is exactly $t^3+\frac{3rt}{2}$ (or $t^3+\frac{3rt-1}{2}$ if $t$ and $r$ are both odd ) and similarly $f(t^2-r)$ is exactly $t^3-\frac{3rt}{2}$ (or $t^3-\frac{3rt+1}{2}$ if $t$ and $r$ are both odd ). So we already know that the answer is "yes" for most of the numbers.

Answer 1

Here is a proof that $|g(n)|\le 1$ for all but finitely many $n$. You can extract an explicit bound for $n$ from the argument and check the smaller values by hand.

If $f(n)=n^{3/2}$ without the floor, then $g(n)\sim \frac{3}{4\sqrt n}$, so it is positive and tends to 0. When you replace $n^{3/2}$ by its floor, $g(n)$ changes by at most 2, hence the only chance for failure is to have $g(n)=2$ when the fractional parts of $n^{3/2}$ and $(n+2)^{3/2}$ are very small and the fractional part of $(n+1)^{3/2}$ is very close to 1 (the difference is less than $\frac{const}{\sqrt{n}}$).

Let $a,b,c$ denote the nearest integers to $n^{3/2}$, $(n+1)^{3/2}$ and $(n+2)^{3/2}$. Then $c-2b+a=0$ because it is an integer very close to $(n+2)^{3/2}-2(n+1)^{3/2}+n^{3/2}$. Denote $m=c-b=b-a$. Then $(n+1)^{3/2}-n^{3/2}<m$ and $(n+2)^{3/2}-(n+1)^{3/2}>m$. Observe that $$ \frac{3}{2}\sqrt{n}<(n+1)^{3/2} - n^{3/2} < \frac{3}{2}\sqrt{n+1} $$ (the bounds are just the bounds for the derivative of $x^{3/2}$ on $[n,n+1]$. Therefore $$ \frac{3}{2}\sqrt{n} < m < \frac{3}{2}\sqrt{n+2} $$ or, equivalently, $$ n < \frac49 m^2 < n+2. $$ If $m$ is a multiple of 3, this inequality implies that $n+1=\frac49 m^2=(\frac23m)^2$, then $(n+1)^{3/2}=(\frac23m)^3$ is an integer and not slightly smaller than an integer as it should be. If $m$ is not divisible by 3, then $$ n+1 = \frac49 m^2 + r $$ where $r$ is a fraction with denominator 9 and $|r|<1$. From Taylor expansion $$ f(x+r) = f(x) +r f'(x) +\frac12 r^2 f''(x+r_1), \ \ 0<r_1<r, $$ for $f(x)=x^{3/2}$, we have $$ (n+1)^{3/2} = (\frac49 m^2 + r)^{3/2} = \frac8{27}m^3 + mr + \delta $$ where $0<\delta<\frac1{4m}$. This cannot be close to an integer because it is close (from above) to a fraction with denominator 27.

Answer 2

It's not too hard to put a bound on the size of second differences (since without the truncation, they are bounded above by a constant times $n^{-1/2}$), but getting the bound down to one seems delicate. It looks like it can be done with mindless brute force, though. I won't write all of the cases, but here is a start. Write $n = t^2 + r$, for integers $t,r$ satisfying $|r| \leq t$. The binomial theorem implies $n^{3/2} = t^3 + (3/2)tr + (3/8)r^2/t - (1/16)r^3/t^3 + (3/128)r^4/t^5 - \dots$. I'll look at the case where $t$ is even. Then

• $f(n) = t^3 + (3/2)tr + \lfloor (3/8)r^2/t - (1/16)r^3/t^3 + (3/128)r^4/t^5 - \dots \rfloor$
• $f(n+1) = t^3 + (3/2)t(r+1) + \lfloor (3/8)(r+1)^2/t - (1/16)(r+1)^3/t^3 + (3/128)(r+1)^4/t^5 - \dots \rfloor$
• $f(n+2) = t^3 + (3/2)t(r+2) + \lfloor (3/8)(r+2)^2/t - (1/16)(r+2)^3/t^3 + (3/128)(r+2)^4/t^5 - \dots \rfloor$.

$g(n)$ then has no contributions from the first two terms of each series, since they cancel. Therefore:

$g(n) = \lfloor (3/8)r^2/t - (1/16)r^3/t^3 + (3/128)(r+1)^4/t^5 - \dots \rfloor +$ $\qquad + 2\lfloor (3/8)r^2/t + (3/4)r/t - (1/16)(r+1)^3/t^3 + (3/8t) + (3/128)(r+2)^4/t^5 - \dots \rfloor +$ $\qquad + \lfloor (3/8)r^2/t + (3/2)r/t - (1/16)(r+2)^3/t^3 + (3/2t) + (3/128)(r+2)^4/t^5 - \dots \rfloor$.

At this point, you can break the analysis into more cases involving the fractional part of $(3/8)r^2/t$ and the size of $r/t$, and then invoke some estimates about the remaining parts of the sum.

Answer 3

For $-1<h<1$, $$(1+h)^{3/2}+(1-h)^{3/2} = 2 \sum_{n=0}^{+\infty}{3/2 \choose 2n}h^{2n},$$ where $${3/2 \choose 2n} = \prod_{k=1}^ {2n} \frac{5/2-k}{k}.$$ For $n \ge 1$, since $(-1)^{2n-2}=1$, we get $${3/2 \choose 2n} = \frac{3}{8}\prod_{k=3}^{2n} \frac{5/2-k}{k} = \frac{3}{8}\prod_{k=3}^{2n} \frac{k-5/2}{k} \in ~\Big[0,\frac{3}{8}\Big].$$ Thus, for $-1<h<1$, $$2 \le (1+h)^{3/2}+(1-h)^{3/2} < 2 + \frac{3}{4}\sum_{n=0}^{+\infty}{3/2 \choose 2n}h^{2n} = 2 + \frac{3}{4}\frac{h^2}{1-h^2}.$$ Given $x>1$, one can apply these inequalities with $h=x^{-1}$.
$$2 \le (1+x^{-1})^{3/2}+(1-x^{-1})^{3/2} < 2 + \frac{3}{4}\frac{x^{-2}}{1-x^{-2}} = 2 + \frac{3}{4}\frac{1}{x^2-1}.$$ Multiplying by $x^{3/2}$ yields $$2x^{3/2} \le (x+1)^{3/2}+(x-1)^{3/2} < 2x^{3/2} + \frac{3}{4}\frac{1}{x^{1/2}-x^{-3/2}}.$$ The quantity $x^{1/2}-x^{-3/2}$ increases when $x$ increases.

Thus, when $x \ge 2$, $$x^{1/2}-x^{-3/2} \ge 2^{1/2}-2^{-3/2} = \frac{3}{4}2^{1/2},$$ so $$2x^{3/2} \le (x+1)^{3/2}+(x-1)^{3/2} < 2x^{3/2} + 2^{-1/2} < 2x^{3/2}+1.$$ As a result, $$2 \lfloor x^{3/2} \rfloor \le 2x^{3/2} \le (x+1)^{3/2}+(x-1)^{3/2} < \lfloor (x+1)^{3/2} \rfloor + \lfloor (x-1)^{3/2} \rfloor + 2.$$ and $$\lfloor (x+1)^{3/2} \rfloor + \lfloor (x-1)^{3/2} \rfloor \le (x+1)^{3/2}+(x-1)^{3/2} < 2x^{3/2}+1 < 2 \lfloor x^{3/2} \rfloor + 2,$$ so the integer $\lfloor (x+1)^{3/2} \rfloor + \lfloor (x-1)^{3/2} \rfloor - 2 \lfloor x^{3/2} \rfloor$ is in $\{-1,0,1\}$.