Geometric Block Exponents and a Uniform Mixed-Alphabet Sumset Inequality

Abstract

We study sharp exponents in inequalities for pairs of finite geometric blocks. We characterize exactly when the endpoint $t=1$ determines the optimal exponent and compute the exponent that is uniform in the length of one block. For a two-term first block the answer is $p_0=\log 4/\log 6$. This yields a uniform two-slice max-convolution inequality and, for every $m,d\ge1$, the dimension-free mixed-alphabet sumset bound \[ |A+B|\ge (|A||B|)^{p_0}, \qquad A\subset\{0,1\}^d,\quad B\subset\{0,1,\ldots,m\}^d. \] For every $m\ge2$, the exponent $p_0$ is best possible; for $m=1$, a larger exponent is available.