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Let $f(x)$ be a polynomial with real coefficients, and let $||\cdot||$ be the distance-from-the-nearest-integer function. It is known that for any $\epsilon > 0$, the set $S$ of positive integer solutions of the inequality $||f(x)|| < \epsilon$ has bounded gaps. This means that if $x_1 < x_2 < \ldots$ are the elements of $S$ in increasing order, then the set of differences $x_{n+1}-x_n$ is bounded. This is proved in "Simultaneous Diophantine Approximation and IP Sets" by Furstenberg and Weiss, Acta Arith. 1988.

I would like to know how to compute the maximum gap, i.e. the maximum of the values $x_{n+1}- x_n$, given a specific polynomial and a specific value of $\epsilon$. I can do this for linear polynomials, but not for quadratics. The proof of Furstenberg and Weiss appears to be non-constructive.

To take a particular example, what is the maximum possible gap between successive solutions of the inequality $||2^{1/2}x^2|| < .01$? Can anyone suggest a method, no matter how impractical, that would eventually lead to an answer? Could it be that, even in this particular case, no one knows how to find the maximum gap?

Note: Experimentation suggests that the maximum gap is 627, which occurs for the first time following the solution $x=1115714$.

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Let $f(x)$ be a polynomial with real coefficients, and let $||\cdot||$ be the distance-from-the-nearest-integer function. It is known that for any $\epsilon > 0$, the set $S$ of positive integer solutions of the inequality $||f(x)|| < \epsilon$ has bounded gaps. This means that if $x_1 < x_2 < \ldots$ are the elements of $S$ in increasing order, then the set of differences $x_{n+1}-x_n$ is bounded. This is proved in "Simultaneous Diophantine Approximation and IP Sets" by Furstenberg and Weiss, Acta Arith. 1988.

I would like to know how to compute the maximum gap, i.e. the maximum of the values $x_{n+1}- x_n$, given a specific polynomial and a specific value of $\epsilon$. I can do this for linear polynomials, but not for quadratics. The proof of Furstenberg and Weiss appears to be non-constructive.

To take a particular example, what is the maximum possible gap between successive solutions of the inequality $||2^{1/2}x^2|| < .01$? Can anyone suggest a method, no matter how impractical, that would eventually lead to an answer? Could it be that, even in this particular case, no one knows how to find the maximum gap?

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Let $f(x)$ be a polynomial with real coefficients, and let $||\cdot||$ be the distance-from-the-nearest-integer function. It is known that for any $\epsilon > 0$, the set $S$ of positive integer solutions of the inequality $||f(x)|| < \epsilon$ has bounded gaps. This means that if $x_1 < x_2 < \ldots$ are the elements of $S$ in increasing order, then the set of differences $x_{n+1}-x_n$ is bounded. This is proved in "Simultaneous Diophantine Approximation and IP Sets" by Furstenberg and Weiss, Acta Arith. 1988.

I would like to know how to compute the maximum gap, i.e. the maximum of the values $\max_n x_{n+1}- x_n$, given a specific polynomial and a specific value of epsilon. $\epsilon$. I can do this for linear polynomials, but not for quadratics. The proof of Furstenberg and Weiss appears to be non-constructive.

To take a particular example, what is the maximum possible gap between successive solutions of the inequality $||2^{1/2}x^2|| < .01$? Can anyone suggest a method, no matter how impractical, that would eventually lead to an answer?

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