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Suppose you can choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+ n})$$f({\alpha_+, n})$ positive and at least $f({\alpha_- n})$$f({\alpha_-, n})$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choice for functions $f(a,b)=2^{ab}$ and $f(a,b)=2^{(\log b)^{1+a}}$?

Suppose you can choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+ n})$ positive and at least $f({\alpha_- n})$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choice for functions $f(a,b)=2^{ab}$ and $f(a,b)=2^{(\log b)^{1+a}}$?

Suppose you choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+, n})$ positive and at least $f({\alpha_-, n})$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choice for functions $f(a,b)=2^{ab}$ and $f(a,b)=2^{(\log b)^{1+a}}$?

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user76479
user76479

Suppose you can choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+ n})$ positive and at least $f({\alpha_- n})$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choice for functions $f(a,b)=2^{ab}$ and $f(a,b)=2^{(\log b)^{1+a}}$?

I think there is no difference in both choices of $f$.

Suppose you can choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+ n})$ positive and at least $f({\alpha_- n})$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choice for functions $f(a,b)=2^{ab}$ and $f(a,b)=2^{(\log b)^{1+a}}$?

I think there is no difference in both choices of $f$.

Suppose you can choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+ n})$ positive and at least $f({\alpha_- n})$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choice for functions $f(a,b)=2^{ab}$ and $f(a,b)=2^{(\log b)^{1+a}}$?

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user76479
user76479

Suppose you can choose $n$ distinct random numbers from a contiguous subset of cardinality $2^{\beta n}$$f({\beta, n})$ with at least $2^{\alpha_+ n}$$f({\alpha_+ n})$ positive and at least $2^{\alpha_- n}$$f({\alpha_- n})$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choiceschoice for functions $f(a,b)=2^{ab}$ and $f(a,b)=2^{(\log b)^{1+a}}$?

I think there is no difference in both choices of $f$.

Suppose you can choose $n$ distinct random numbers from a contiguous subset of cardinality $2^{\beta n}$ with at least $2^{\alpha_+ n}$ positive and at least $2^{\alpha_- n}$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choices?

Suppose you can choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+ n})$ positive and at least $f({\alpha_- n})$ negative values from a set of $2^{n+1}+1$ integers from $\{-2^n,\dots,-1,0,1,\dots,2^n\}$ at a fixed $\alpha_+,\alpha_-,\beta\in(0,1)$ where $\alpha_+,\alpha_-<\beta$ holds what is the maximum number and average number of subsets that sum to $0$ among all such choice for functions $f(a,b)=2^{ab}$ and $f(a,b)=2^{(\log b)^{1+a}}$?

I think there is no difference in both choices of $f$.

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user76479