find $a,b,c$ such that $ \big| a + b\sqrt{2}+ c\sqrt{3} \big| <10^{-3} $

Question

i know that linear combinations of $1$, $\sqrt{2}$, and $\sqrt{3}$ are dense in the real number line.

$$ \overline{ \mathbb{Z}[\sqrt{2},\sqrt{3}] } = \mathbb{R} $$

how quickly to these numbers converge to the entire number line. what are the smallest $a,b,c$ we can find so that

$$ \big| a + b\sqrt{2}+ c\sqrt{3} \big| <10^{-3} \text{ or }10^{-6} $$

by the pigeonhole principles we can always find $-N<a,b,c<N $ such that

$$ \big| a + b\sqrt{2}+ c\sqrt{3} \big| <N^{-2} $$

Answer 1

Here are some lower bounds for the best you can do.

If $\alpha=a+b\sqrt{2}+c\sqrt{3}$ and $|\alpha|<1$ with $a,b,c$ integers, and $abc\neq 0$, then $$|\alpha|>\frac{1}{(2|b|\sqrt{2}+1)(2|c|\sqrt{3}+1)(2|a|+1)}\tag{1}$$

I get this lower bound since: $\alpha(\alpha-2b\sqrt{2})(\alpha-2c\sqrt{3})(\alpha-2a)$ must be an integer.

If $a=0$, the same trick has that if $|\alpha|<1$ then $|\alpha|>\frac{1}{2|c|\sqrt{3}+1}$.

If $b=0$ or $c=0$, the best you can do is with the continued fractions of $\sqrt{3}$ and $\sqrt{2}$, respectively.

You get (1) in general, for all $a,b,c$.

You tend to get some "good" values when:

$$N(a,b,c)=|\alpha(\alpha-2b\sqrt{2})(\alpha-2c\sqrt{3})(\alpha-2a)|=|a^4+4b^4+9c^4-(4a^2b^2+6a^2c^2+12c^2b^2)|$$

is a small integer.

For example, $N(14,5,4)=4$ and $14-5\sqrt{2}-4\sqrt{3}$ is a good approximation.

This is because if $|\alpha|<1$ then:

$$\frac{N(a,b,c)}{(2|a|+1)(2|b|\sqrt{2}+1)(2|c|\sqrt{3}+1)}<|\alpha|<\frac{N(a,b,c)}{(2|a|-1)(2|b|\sqrt{2}-1)(2|c|\sqrt{3}-1)}$$

Answer 2

Best for $1/1000$ then $1/1000000$

  23 was |a| + |b| + |c|     a  14  b  -5  c  -4 numb 0.0007289578590154999
  64 was |a| + |b| + |c|     a  1  b  -35  c  28 numb -5.207112976868267e-05
  75 was |a| + |b| + |c|     a  13  b  30  c  -32 numb 0.0007810289887841826
  79 was |a| + |b| + |c|     a  15  b  -40  c  24 numb 0.0006768867292468173

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 911 was |a| + |b| + |c|     a  28  b  -495  c  388 numb -3.795774605741542e-08

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Best for $10^{-8}$

   11739 was |a| + |b| + |c|     a  817  b  5753  c  -5169 numb 8.890160607677444e-09
   87678 was |a| + |b| + |c|     a  1103  b  -48011  c  38564 numb -8.490950165196409e-09
   77573 was |a| + |b| + |c|     a  1920  b  -42258  c  33395 numb 3.992077779457759e-10
   67468 was |a| + |b| + |c|     a  2737  b  -36505  c  28226 numb 9.28936927380164e-09

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Best for $10^{-9},$ in both the $L^1$ and $L^\infty$ norms, is the first line

   48677 was |a| + |b| + |c|; a  17841  b  11305  c  -19531 numb 1.001581040327437e-10
   62917 was |a| + |b| + |c|; a  37602  b  -19648  c  -5667 numb 5.995275387249421e-10
   64578 was |a| + |b| + |c|; a  19761  b  -30953  c  13864 numb 4.993694346921984e-10
   77573 was |a| + |b| + |c|; a  1920  b  -42258  c  33395 numb 3.992077779457759e-10
   97354 was |a| + |b| + |c|; a  35682  b  22610  c  -39062 numb 2.003162080654874e-10
  122410 was |a| + |b| + |c|; a  15921  b  53563  c  -52926 numb -2.990461211993534e-10
  129156 was |a| + |b| + |c|; a  39522  b  -61906  c  27728 numb 9.987388693843968e-10
  142151 was |a| + |b| + |c|; a  21681  b  -73211  c  47259 numb 8.985736599242955e-10
  171087 was |a| + |b| + |c|; a  33762  b  64868  c  -72457 numb -1.988880171666096e-10
  219764 was |a| + |b| + |c|; a  51603  b  76173  c  -91988 numb -9.872280770650832e-11

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