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#3: Post edited by user avatar TextKit‭ · 2022-05-14T23:09:16Z (almost 2 years ago)
  • ## My questions
  • 1. Can you cite or scan the question from the source? I am leery, because the claim appears false.
  • 2. Is g(x) supposed to be convex?
  • ## Game plan
  • I shall construct, on top of $f(x)=x$, a function which occasionally jumps up a constant amount over littler and littler intervals. This means I can create a function with the growth rate of $x$, but with as much derivative growth as I want.
  • ## Counter Example
  • Let $h(x)$ be any nonnegative continuous function with the property that $h(0) = 1$ and $h \le 1$ everywhere, and $h = 0
  • $ outside the interval $[-0.25,0.25]$. Let $H(x)= h(x) + 2h(2(x-1)) + 4h(4(x-2)) + 8h((x-3)/8) + ...$
  • Then $H(x)$ converges to a continuous function, because the supports of each summand are disjoint. Observe that $H(n) = 2n$.
  • Let $g(x) = x + \int^0_x H(t) \, dt$. Then certainly $g$ is $C^1$, increasing, and tends to infinity.
  • I constructed $h \le 1$, and $0$ outside a finite interval. Thus $ \int^0_x ah(at) = \int^0_{x/a} h(u) du \le C$ for some constant C. Then $g(x) \le x + C(x+1)$, since at most $x+1$ of the rescaled $h's
  • eq 0$.
  • So $g^{1+\epsilon}$ grows at most polynomially. But we just constructed $g$ so that $g' \ge H$ has an exponentially growing SUBsequence!
  • ## My questions
  • 1. Can you cite or scan the question from the source? I am leery, because the claim appears false.
  • 2. Is g(x) supposed to be convex?
  • ## Game plan
  • I shall construct, on top of $f(x)=x$, a function which occasionally jumps up a constant amount over littler and littler intervals. This means I can create a function with the growth rate of $x$, but with as much derivative growth as I want.
  • ## Counter Example
  • Let $h(x)$ be any nonnegative continuous function with the property that $h(0) = 1$ and $h \le 1$ everywhere, and $h = 0
  • $ outside the finite interval $[-0.25,0.25]$. Let $H(x)= h(x) + 2h(2(x-1)) + 4h(4(x-2)) + 8h((x-3)/8) + ...$
  • Then $H(x)$ converges to a continuous function, because the supports of each summand are disjoint. Observe that $H(n) = 2n$.
  • Let $g(x) = x + \int^0_x H(t) \, dt$. Then certainly $g$ is $C^1$, increasing, and tends to infinity.
  • I constructed $h \le 1$, and $h = 0$. Thus $ \int^0_x ah(at) = \int^0_{x/a} h(u) du \le C$ for some constant C. Then $g(x) \le x + C(x+1)$, since at most $x+1$ of the rescaled $h's
  • eq 0$.
  • So $g^{1+\epsilon}$ grows at most polynomially. But we just constructed $g$ so that $g' \ge H$ has an exponentially growing SUBsequence!
#2: Post edited by user avatar TextKit‭ · 2022-05-14T23:08:05Z (almost 2 years ago)
  • ## My questions
  • Can you scan the question as you were assigned? I am leery, because the claim appears false.
  • Is g(x) supposed to be convex?
  • ## Game plan
  • I shall construct, on top of $f(x)=x$, a function which occasionally jumps up a constant amount over littler and littler intervals. This means I can create a function with the growth rate of $x$, but with as much derivative growth as I want.
  • ## Counter Example
  • Let $h(x)$ be any nonnegative continuous function with the property that $h(0) = 1$ and $h \le 1$ everywhere, and $h = 0
  • $ outside the interval $[-0.25,0.25]$. Let $H(x)= h(x) + 2h(2(x-1)) + 4h(4(x-2)) + 8h((x-3)/8) + ...$
  • Then $H(x)$ converges to a continuous function, because the supports of each summand are disjoint. Observe that $H(n) = 2n$.
  • Let $g(x) = x + \int^0_x H(t) \, dt$. Then certainly $g$ is $C^1$, increasing, and tends to infinity.
  • I constructed $h \le 1$, and $0$ outside a finite interval. Thus $ \int^0_x ah(at) = \int^0_{x/a} h(u) du \le C$ for some constant C. Then $g(x) \le x + C(x+1)$, since at most $x+1$ of the rescaled $h's \neq 0$.
  • So $g^{1+\epsilon}$ grows at most polynomially. But we just constructed $g$ so that $g' \ge H$ has an exponentially growing SUBsequence!
  • ## My questions
  • 1. Can you cite or scan the question from the source? I am leery, because the claim appears false.
  • 2. Is g(x) supposed to be convex?
  • ## Game plan
  • I shall construct, on top of $f(x)=x$, a function which occasionally jumps up a constant amount over littler and littler intervals. This means I can create a function with the growth rate of $x$, but with as much derivative growth as I want.
  • ## Counter Example
  • Let $h(x)$ be any nonnegative continuous function with the property that $h(0) = 1$ and $h \le 1$ everywhere, and $h = 0
  • $ outside the interval $[-0.25,0.25]$. Let $H(x)= h(x) + 2h(2(x-1)) + 4h(4(x-2)) + 8h((x-3)/8) + ...$
  • Then $H(x)$ converges to a continuous function, because the supports of each summand are disjoint. Observe that $H(n) = 2n$.
  • Let $g(x) = x + \int^0_x H(t) \, dt$. Then certainly $g$ is $C^1$, increasing, and tends to infinity.
  • I constructed $h \le 1$, and $0$ outside a finite interval. Thus $ \int^0_x ah(at) = \int^0_{x/a} h(u) du \le C$ for some constant C. Then $g(x) \le x + C(x+1)$, since at most $x+1$ of the rescaled $h's \neq 0$.
  • So $g^{1+\epsilon}$ grows at most polynomially. But we just constructed $g$ so that $g' \ge H$ has an exponentially growing SUBsequence!
#1: Initial revision by user avatar TextKit‭ · 2022-05-14T23:07:32Z (almost 2 years ago)
## My questions

Can you scan the question as you were assigned? I am leery, because the claim appears false.

Is g(x) supposed to be convex? 

## Game plan


I shall construct, on top of $f(x)=x$, a function which occasionally jumps up a constant amount over littler and littler intervals. This means I can create a function with the growth rate of $x$, but with as much derivative growth as I want.


## Counter Example

 Let $h(x)$ be any nonnegative continuous function with the property that  $h(0) = 1$ and $h \le 1$ everywhere, and $h = 0
$ outside the interval $[-0.25,0.25]$. Let $H(x)= h(x) + 2h(2(x-1)) + 4h(4(x-2)) + 8h((x-3)/8) + ...$ 

Then $H(x)$ converges to a continuous function, because the supports of each summand are disjoint. Observe that $H(n) = 2n$.

Let $g(x) = x + \int^0_x H(t) \, dt$. Then certainly $g$ is $C^1$, increasing, and tends to infinity. 

I constructed $h \le 1$, and $0$ outside a finite interval. Thus $ \int^0_x  ah(at) = \int^0_{x/a} h(u) du \le C$ for some constant C. Then $g(x) \le x + C(x+1)$, since at most $x+1$ of the rescaled $h's \neq 0$.

So $g^{1+\epsilon}$ grows at most polynomially. But we just constructed $g$ so that $g' \ge H$ has an exponentially growing SUBsequence!