Bounds for generalized Sidon sets

Let ΓΓ be an abelian group and g≥h≥2g≥h≥2 be integers. A set A⊂ΓA⊂Γ is a Ch[g]Ch[g]-set if given any set X⊂ΓX⊂Γ with |X|=h|X|=h, and any set {k1,…,kg}⊂Γ{k1,…,kg}⊂Γ, at least one of the translates X+kiX+ki is not contained in AA. For any g≥h≥2g≥h≥2, we prove that if A⊂{1,2,…,n}A⊂{1,2,…,n} is a Ch[g]Ch[g]-set in ZZ, then |A|≤(g−1)1/hn1−1/h+O(n1/2−1/2h)|A|≤(g−1)1/hn1−1/h+O(n1/2−1/2h). We show that for any integer n≥1n≥1, there is a C3[3]C3[3]-set A⊂{1,2,…,n}A⊂{1,2,…,n} with |A|≥(4−2/3+o(1))n2/3|A|≥(4−2/3+o(1))n2/3. We also show that for any odd prime pp, there is a C3[3]C3[3]-set A⊂Fp3 with |A|≥p2−p|A|≥p2−p, which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer h≥3h≥3, there is a Ch[h!+1]Ch[h!+1]-set A⊂{1,2,…,n}A⊂{1,2,…,n} with |A|≥(ch+o(1))n1−1/h|A|≥(ch+o(1))n1−1/h. A set AA is a weak Ch[g]Ch[g]-set if we add the condition that the translates X+k1,…,X+kgX+k1,…,X+kg are all pairwise disjoint. We use the probabilistic method to construct weak Ch[g]Ch[g]-sets in {1,2,…,n}{1,2,…,n} for any g≥h≥2g≥h≥2. Lastly we obtain upper bounds on infinite Ch[g]Ch[g]-sequences. We prove that for any infinite Ch[g]Ch[g]-sequence A⊂NA⊂N, we have A(n)=O(n1−1/h(logn)−1/h)A(n)=O(n1−1/h(logn)−1/h) for infinitely many nn, where A(n)=|A∩{1,2,…,n}|A(n)=|A∩{1,2,…,n}|.