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Let $V \subseteq \mathbb{R}^n$ be a set cut out by a system of finitely many polynomial equations and inequalities with integer coefficients. Let $W$ be the set of all points in the box $[0,1]^n$ that are congruent mod 1 to some point of $V$. I would like to know whether or not there is an algorithm for determining if a given rational number is equal to, greater than or less than the Lebesgue measure of $W$. There may not be one, in light of this argument, which proves, in effect, that there is no algorithm to determine if $W=[0,1]^n$.

Note that one can effectively produce arbitrarily accurate lower bounds on the measure of $W$, by breaking up $V$ into small pieces and translating more and more of these pieces to $[0,1]^n$. The problem is, I don't see how to get arbitrarily accurate upper bounds on the measure of $W$.

Let $V \subseteq \mathbb{R}^n$ be a set cut out by a system of finitely many polynomial equations and inequalities with integer coefficients. Let $W$ be the set of all points in the box $[0,1]^n$ that are congruent mod 1 to some point of $V$. I would like to know whether or not there is an algorithm for determining if a given rational number is equal to, greater than or less than the Lebesgue measure of $W$. There may not be one, in light of this argument, which proves, in effect, that there is no algorithm to determine if $W=[0,1]^n$.

Note that one can effectively produce arbitrarily accurate lower bounds on the measure of $W$, by breaking up $V$ into small pieces and translating more and more of these pieces to $[0,1]^n$. The problem is, I don't see how to get arbitrarily accurate upper bounds on the measure of $W$.

Let $V \subseteq \mathbb{R}^n$ be a set cut out by a system of finitely many polynomial equations and inequalities with integer coefficients. Let $W$ be the set of all points in the box $[0,1]^n$ that are congruent mod 1 to some point of $V$. I would like an algorithm for determining whether a given rational number is equal to, greater than or less than the Lebesgue measure of $W$, or I would like to know that no such algorithm exists. The latter seems to be a serious possibility, in light of   this argument.

Note that one can effectively produce arbitrarily accurate lower bounds on the measure of $W$, by breaking up $V$ into small pieces and translating more and more of these pieces to $[0,1]^n$. The problem is, I don't see how to get arbitrarily accurate upper bounds on the measure of $W$.

Let $V \subseteq \mathbb{R}^n$ be a set cut out by a system of finitely many polynomial equations and inequalities with integer coefficients. Let $W$ be the set of all points in the box $[0,1]^n$ that are congruent mod 1 to some point of $V$. I would like to know whether or not there is an algorithm for determining if a given rational number is equal to, greater than or less than the Lebesgue measure of $W$. There may not be one, in light of  this argument, which proves, in effect, that there is no algorithm to determine if $W=[0,1]^n$.

Note that one can effectively produce arbitrarily accurate lower bounds on the measure of $W$, by breaking up $V$ into small pieces and translating more and more of these pieces to $[0,1]^n$. The problem is, I don't see how to get arbitrarily accurate upper bounds on the measure of $W$.