arXiv subjects
Publications and source records attributed to Rafael Tesoro.
We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs.
Let $Γ$ be an abelian group and $g \geq h \geq 2$ be integers. A set $A \subset Γ$ is a $C_h[g]$-set if given any set $X \subset Γ$ with $|X| = k$, and any set $\{ k_1 , \dots , k_g \} \subset Γ$, at least one of the translates $X+ k_i$ is not contained in $A$. For any $g \geq h \geq 2$, we prove that if $A \subset \{1,2, \dots ,n \}$ is a $C_h[g]$-set in $\mathbb{Z}$, then $|A| \leq (g-1)^{1/h} n^{1 - 1/h} + O(n^{1/2 - 1/2h})$. We show that for any integer $n \geq 1$, there is a $C_3 [3]$-set $A \subset \{1,2, \dots , n \}$ with $|A| \geq (4^{-2/3} + o(1)) n^{2/3}$. We also show that for any odd prime $p$, there is a $C_3[3]$-set $A \subset \mathbb{F}_p^3$ with $|A| \geq p^2 - p$, which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer $h \geq 3$, there is a $C_h[h! +1]$-set $A \subset \{1,2, \dots , n \}$ with $|A| \geq ( c_h +o(1))n^{1-1/h}$. A set $A$ is a \emph{weak $C_h[g]$-set} if we add the condition that the translates $X +k_1, \dots , X + k_g$ are all pairwise disjoint. We use the probabilistic method to construct weak $C_h[g]$-sets in $\{1,2, \dots , n \}$ for any $g \geq h \geq 2$. Lastly we obtain upper bounds on infinite $C_h[g]$-sequences. We prove that for any infinite $C_h[g$]-sequence $A \subset \mathbb{N}$, we have $A(n) = O ( n^{1 - 1/h} ( \log n )^{ - 1/h} )$ for infinitely many $n$, where $A(n) = | A \cap \{1,2, \dots , n \}|$.
Consider the sets of integers $A$ that avoid any arrangement of $g$ congruent $h$-subsets. Our findings refine and improve upon some results by Erdős and Harzheim about these sets.
For $h=3$ and $h=4$ we prove the existence of infinite $B_h$ sequences $\B$ with counting function $$\mathcal{B}(x)= x^{\sqrt{(h-1)^2+1}-(h-1) + o(1)}.$$ This result extends a construction of I. Ruzsa for $B_2$ sequences.