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I know that there are many unanswered questions and conjectures when it comes to additive bases (of order $k=2$) of $\mathbb{N}.$ However, for the question I have in mind, I think it should be elementary to prove it, but I am having trouble doing so.

Conjecture: $\ A_x:= \{ \left\lfloor x^n \right\rfloor: n\in\mathbb{N}\ \}\ $ is not an additive basis (of order $k=2$) of $\mathbb{N}$ if $x>1\ ?$

This is probably very easy to show for "large" values of $x$ like $x \geq 1.3:\ $ all you have to do is write a program and find the least integer $q$ that causes failure, i.e. there does not exist $p_1,\ p_2 \in A_x\ $ such that $p_1 + p_2 = q.$ But how to show this for all $x>1,$ in particular, like for $x= 1.00000001 ?\ $

I imagine the proof should include the fact that, given $ j,\ \exists\ N\ $ such that $\left\lfloor x^{n+1} \right\rfloor - \left\lfloor x^{n} \right\rfloor > j\ \forall\ n\geq N.\ $ Or perhaps we need to use some stronger statement? Or is this problem not solved? I feel like it should be solvable with elementary methods, using the fact that $A_x$ becomes sparse as $n$ gets very large...

Adam Rubinson
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1 Answers1

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Let $X>10$. We find an upper bound of the sets:

$$ [A_x + A_x]_X :=\{ n+m\leq X \ | \ n,m\in A_x \},$$

$$ [A_x]_X:=\{ n\leq X \ | \ n\in A_x \}. $$

The elememts of $[A_x+A_x]_X$ are determined by pairs of elements of $[A_x]_X$, and we have $|[A_x]_X|\leq C \log_x X$ for some absolute constant $C>0$.

A crude upper bound is therefore $|[A_x+A_x]_X|\leq (C\log_x X)^2$.

Then natural numbers up to $X$ represented by a sum of at most $2$ elements of $A_x$ is at most $$ (C\log_x X)^2+ C\log_x X. $$

This is smaller than $X$. Thus, almost all (asymptotic density $1$) positive integers are not the sum of at most $2$ elements of $A_x$.

Sungjin Kim
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