Infinite Generalized Sidon Sets with a Negative Answer to a Problem of Szemerédi

Abstract

We construct infinite $B_h$ sets that repeatedly attain the density scale of finite $B_h$ sets. For every integer $h\geq2$ and every finite $B_h$ set $F$ of positive integers, we obtain an infinite $B_h$ set $A\supseteq F$ such that $$ \limsup_{x\to\infty}\frac{A(x)}{x^{1/h}}=\frac1h, \qquad \liminf_{x\to\infty}\frac{A(x)}{x^{1/h}}=0. $$ The key is to preserve density while preventing additive relations between successive blocks. A positional encoding of the finite-field moment curve has $q$ elements in an interval of length $O_h(q^h)$, while, for each $1\leq r<h$, sums arising from distinct $r$-element multisets are separated by more than $q$. This separation allows whole blocks to be appended to a prescribed finite set without a further dilation. In particular, our result gives a negative answer to a problem posed by Szemerédi \cite{szemeredi}.