Separation of lattice points on the Mordell elliptic curve

Question

Consider the Mordell equation x^3 – y^2 = k, where x is a non-square positive integer and y^2 is the perfect square nearest to x^3. Noam Elkies (see http://www.math.harvard.edu/~elkies/hall.html) found in 1998 that there are only 25 integers below 10^18 for which |k| < sqrt(x) (the first three being 2, 5234 and 8158). For these 25 numbers (and for the approximately 30 larger numbers found to date) the separation is huge.

I am not aware of any research results that prove a minimum separation between such numbers in general: If x is such a number, what is the minimum separation between x and the next highest such number? Is anyone aware of such research and/or able to comment on how simple/difficult this question is? The question obviously also relates to the separation between lattice points on the Mordell elliptic curve for large x (k fixed).

Elkies (Rational points near curves and small nonzero |x^3 − y^2| via lattice reduction, May 2000) has proved an upper bound (order of sqrt(N).log(N)) for the number of such points not exceeding N, but it is not clear to me whether his method implies any minimum separation. I would greatly appreciate any information on the existence or otherwise of research on this question.

Answer 1

Hall's conjecture says that for every $\epsilon>0$ there is a $C_\epsilon$ such that if $x$ and $y$ are integers with $x^3-y^2\ne0$, then $$ |x^3 - y^2| \ge C_\epsilon \max\{|x^3|,|y^2|\}^{1/6-\epsilon}, $$ so this would imply that the separation indeed gets quite large as $x$ and $y$ increase. The polynomial version, i.e., when $x$ and $y$ are in $\mathbb{C}[T]$, was proven by Davenport with $\epsilon=0$, i.e., if $x^3 - y^2 \ne 0$ then $$ \deg(x^3-y^2) \ge \frac16 \max\{\deg(x^3),\deg(y^2)\} + C $$ for an absolute constant $C$ (in fact, we can use $C=1$). More generally in the polynomial (or function field) case, one can easily prove lower bounds for $\deg(x^n-y^m)$, with analogous conjectures over $\mathbb{Z}$ (or over number fields).

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Addendum: As Ryan D'Mello points out, the above doesn't really answer the question, which asks about gaps between $x$ values of solutions to $0<|x^3-y^2|<\sqrt{x}$. However, I think that in order to get a reasonable answer, one will need to assume something like Hall's conjecture. Alternatively, for a given (small) $\epsilon>0$, one might hope to unconditionally prove a gap estimate for the set $$ \bigl\{ x\in\mathbb{Z} : \text{there exists $y\in\mathbb{Z}$ with $|x|^{1/2-\epsilon}<|x^3-y^2|<|x|^{1/2+\epsilon}$} \bigr\}, $$ where the size of the gaps depends on $\epsilon$.