For sequences of $n$ nonnegative integer vectors in $\mathbb Z^k$ with distinct subset sums, we prove, for fixed $k$, the uniform lower bound $M\ge (C_k-o(1))2^{n/k}/\sqrt n$ for the largest coordinate, where \[ C_k=\sqrt{\frac{2k}{\pi}}\, \frac{(k+1)^{(k-1)/(2k)}}{\sqrt{\lfloor (k+1)^2/4\rfloor}}. \] The proof compares a Gaussian-kernel expectation with a sum over a translated lattice, uniformly over the vector sequence. We optimise the resulting quadratic-form constant using a vertex second-moment certificate. We show that $C_k$ also exceeds the variance constant obtained by optimising an ellipsoidal geometry.
For a positive integer $k$, let $f(k)$ be the minimum number of distinct nonempty subset sums of a zero-sum-free $k$-element set, where the minimum is taken over all finite abelian groups. We prove $f(k)\ge \frac{k^2}{2}-13k\log(k+1)-116k$. Together with the classical upper bound, this gives $f(k)=k^2/2+O(k\log k)$, wi...
We resolve a conjecture of \L{}uczak, Mieczkowska and \v{S}ileikis on the maximal tail probability of sums of independent and identically distributed nonnegative random variables. Specifically, for every integer $k\ge1$ and every $0\le x\le1/k$, we prove \[ \sup_{\mathbb E X_1\le x}\mathbb{P}(X_1+\cdots+X_k\ge1) =\max\...
Let $n\ge2$ be a fixed integer, and let $K,x,y$ be positive integers satisfying $2\le K\le y<x$. Consider a polynomial \[ f(u)=\alpha u^n+\alpha_{n-1}u^{n-1}+\cdots+\alpha_1u+\alpha_0 \] with real coefficients. We study the quantity \[ V_n(K,x,y) = \sum_{k=1}^{K} \left| \sum_{x-y<p\le x}e(kf(p)) \right|, \qquad e(t)=e^...
We prove that for every integer $n>2$ the largest good lattice point matrix $U_n=[u_h:h\in H_n]$, whose columns are generated by all integers $h<n$ coprime to $n$, has rank exactly $\varphi(n)/2+1$. The upper bound follows from the pairing identity $u_h+u_{n-h}=(n,\dots,n,2n)^{\top}$. For the lower bound the columns ar...
Let $m(p)$ be the least size of a subset of $\F_p$ with at least two elements for which every sum has two distinct representations as unordered pairs, allowing repetition. We prove that, for every prime $p\ge64$, \[ m(p)\ge2^{-80}\left(\frac{\log p}{\log\log p}\right)^2. \] The argument compresses the full integer coll...
In this note, we extend the matrix-multiplication analysis by OpenAI to rectangular products and prove $\omega(1,k,1)\le2$ for $0\le k\le\frac{1}{2}$, and $\omega(1,k,1)\le1+k+\frac{1}{4k}$ for $k\ge\frac{1}{2}$. In particular, $\omega(1, \frac{1}{2},1)=2$ and the dual exponent satisfies $\alpha\ge\frac{1}{2}$. The pro...
P. Uznański· 5 citations· ⚡2
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