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YCor
YCor
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Is Every Positiveevery positive-Density Set ofdensity set of Positive Integers Almost Rationalpositive integers almost rational?

(All the sets in this discussion are sets of positive integers.) I say a set $V$ is “rational” if it satisfies one of the following conditions:

I. $V$ is finite (orpossibly empty);

II. $V$ can be written as the union of a finite (possibly empty) set and finitely many arithmetic progressions of infinite length.

Letting $\Delta$ denote the symmetric difference operation, I say that a set $W$ is “almost rational” if there exists a rational set $V$ such that $V\Delta W$ has zero natural density. Clearly, every $W$ of density $0$ or $1$ is almost rational (with $V$ being any finite set in the former case, and $V$ being any set whose complement is finite, in the latter case). My question is: are there any sets of positive natural density which are not almost rational?

Is Every Positive-Density Set of of Positive Integers Almost Rational?

(All the sets in this discussion are sets of positive integers.) I say a set $V$ is “rational” if it satisfies one of the following conditions:

I. $V$ is finite (or empty);

II. $V$ can be written as the union of a finite (possibly empty) set and finitely many arithmetic progressions of infinite length.

Letting $\Delta$ denote the symmetric difference operation, I say that a set $W$ is “almost rational” if there exists a rational set $V$ such that $V\Delta W$ has zero natural density. Clearly, every $W$ of density $0$ or $1$ is almost rational (with $V$ being any finite set in the former case, and $V$ being any set whose complement is finite, in the latter case). My question is: are there any sets of positive natural density which are not almost rational?

Is every positive-density set of positive integers almost rational?

(All the sets in this discussion are sets of positive integers.) I say a set $V$ is “rational” if it satisfies one of the following conditions:

I. $V$ is finite (possibly empty);

II. $V$ can be written as the union of a finite (possibly empty) set and finitely many arithmetic progressions of infinite length.

Letting $\Delta$ denote the symmetric difference operation, I say that a set $W$ is “almost rational” if there exists a rational set $V$ such that $V\Delta W$ has zero natural density. Clearly, every $W$ of density $0$ or $1$ is almost rational (with $V$ being any finite set in the former case, and $V$ being any set whose complement is finite, in the latter case). My question is: are there any sets of positive natural density which are not almost rational?

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MCS
MCS
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Is Every Positive-Density Set of of Positive Integers Almost Rational?

(All the sets in this discussion are sets of positive integers.) I say a set $V$ is “rational” if it satisfies one of the following conditions:

I. $V$ is finite (or empty);

II. $V$ can be written as the union of a finite (possibly empty) set and finitely many arithmetic progressions of infinite length.

Letting $\Delta$ denote the symmetric difference operation, I say that a set $W$ is “almost rational” if there exists a rational set $V$ such that $V\Delta W$ has zero natural density. Clearly, every $W$ of density $0$ or $1$ is almost rational (with $V$ being any finite set in the former case, and $V$ being any set whose complement is finite, in the latter case). My question is: are there any sets of positive natural density which are not almost rational?