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Let $\mathbb{G}$ be $\mathbb Z_p$ (Integers modulo a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$. For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\epsilon$. Assuming $X$ and $Z$ are disjoint, I'd like to express $\operatorname{dist}_{TV}(Y,U)$ in terms of $\epsilon$, where $U$ is a variable distributed uniformly over $\mathbb G$ and TV is total variation distance, that is $$\operatorname{dist}_{TV}(A,B)=\max_D\left|\Pr(A\in D)-\Pr(B\in D) \right|$$

Something similar was done by Prof. Terrence Tao but with entropy-based distances: https://arxiv.org/pdf/2306.13403.pdf Proposition 1.3.

I wish to find, if possible, an upper bound for $\operatorname{dist}_{TV}(Y,U)$ that does not depend on $|\mathbb G|$ or the size of the support of $X,Y,Z.$

Thank you very much to anyone willing to help!

Let $\mathbb{G} = \mathbb{Z}/p\mathbb{Z}$ (where $p$ is a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$. For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\epsilon$. Assuming $X$ and $Z$ are disjoint, I'd like to express $\operatorname{dist}_{TV}(Y,U)$ in terms of $\epsilon$, where $U$ is a variable distributed uniformly over $\mathbb G$ and TV is total variation distance, that is $$\operatorname{dist}_{TV}(A,B)=\max_D\left|\Pr(A\in D)-\Pr(B\in D) \right|$$

Something similar was done by Prof. Terrence Tao but with entropy-based distances: see Proposition 1.3 of https://arxiv.org/abs/2306.13403.

I wish to find, if possible, an upper bound for $\operatorname{dist}_{TV}(Y,U)$ that does not depend on $|\mathbb G|$ or the size of the support of $X,Y,Z.$

Thank you very much to anyone willing to help!

Let $\mathbb{G}$ be $\mathbb Z_p$ (Integers modulu a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$. For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\epsilon$. Assuming $X$ and $Z$ are disjoint, I'd like to express $\operatorname{dist}_{TV}(Y,U)$ in terms of $\epsilon$, where $U$ is a variable distributed uniformly over $\mathbb G$ and TV is total variation distance, that is $$\operatorname{dist}_{TV}(A,B)=\max_D\left|\Pr(A\in D)-\Pr(B\in D) \right|$$

Something similar was done by Prof. Terrence Tao but with entropy-based distances: https://arxiv.org/pdf/2306.13403.pdf Proposition 1.3.

I wish to find, if possible, an upper bound for $\operatorname{dist}_{TV}(Y,U)$ that does not depend on $|\mathbb G|$ or the size of the support of $X,Y,Z.$

Thank you very much to anyone willing to help!

Let $\mathbb{G}$ be $\mathbb Z_p$ (Integers modulo a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$. For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\epsilon$. Assuming $X$ and $Z$ are disjoint, I'd like to express $\operatorname{dist}_{TV}(Y,U)$ in terms of $\epsilon$, where $U$ is a variable distributed uniformly over $\mathbb G$ and TV is total variation distance, that is $$\operatorname{dist}_{TV}(A,B)=\max_D\left|\Pr(A\in D)-\Pr(B\in D) \right|$$

Something similar was done by Prof. Terrence Tao but with entropy-based distances: https://arxiv.org/pdf/2306.13403.pdf Proposition 1.3.

I wish to find, if possible, an upper bound for $\operatorname{dist}_{TV}(Y,U)$ that does not depend on $|\mathbb G|$ or the size of the support of $X,Y,Z.$

Thank you very much to anyone willing to help!

Let $\mathbb{G}$ be $\mathbb Z_p$. Let $X,Y,Z$ be independent random variables in $\mathbb G$. For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\epsilon$. Assuming $X$ and $Z$ are disjoint, I'd like to express $\operatorname{dist}_{TV}(Y,U)$ in terms of $\epsilon$, where $U$ is a variable distributed uniformly over $\mathbb G$ and TV is total variation distance, that is $$\operatorname{dist}_{TV}(A,B)=\max_D\left|\Pr(A\in D)-\Pr(B\in D) \right|$$

Something similar was done by Prof. Terrence Tao but with entropy-based distances: https://arxiv.org/pdf/2306.13403.pdf Proposition 1.3.

I wish to find, if possible, an upper bound for $\operatorname{dist}_{TV}(Y,U)$ that does not depend on $|\mathbb G|$ or the size of the support of $X,Y,Z.$

Thank you very much to anyone willing to help!

Let $\mathbb{G}$ be $\mathbb Z_p$ (Integers modulu a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$. For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\epsilon$. Assuming $X$ and $Z$ are disjoint, I'd like to express $\operatorname{dist}_{TV}(Y,U)$ in terms of $\epsilon$, where $U$ is a variable distributed uniformly over $\mathbb G$ and TV is total variation distance, that is $$\operatorname{dist}_{TV}(A,B)=\max_D\left|\Pr(A\in D)-\Pr(B\in D) \right|$$

Something similar was done by Prof. Terrence Tao but with entropy-based distances: https://arxiv.org/pdf/2306.13403.pdf Proposition 1.3.

I wish to find, if possible, an upper bound for $\operatorname{dist}_{TV}(Y,U)$ that does not depend on $|\mathbb G|$ or the size of the support of $X,Y,Z.$

Thank you very much to anyone willing to help!