The author investigates the following problem. Let \(A\) denote any set of distinct natural numbers all \(\leq \ell\). For any given set \(M\) of integers, determine \(f(\ell,M)\) defined as the maximum cardinality of such a set \(A\) which contains no subset \(B\subseteq A\) such that \(\sum_{b\in B}b\in M\). Three cases are considered:
(1) \(M=M_ 2\), the set of all squares. Erdős proved that \(f(\ell,M_ 2)\geq 2^{1/3}\ell^{1/3}(1+o(1))\) and the upper bound \(f(\ell,M_ 2)=O(\ell /\log \ell)\) is due to Alon. The author improves this to \(f(\ell,M_ 2)=O(\ell^{4/5+\epsilon}).\)
(2) \(M=M_ r\), the set of all rth powers. Freiman conjectured that for all \(r\geq 2,\)
\[ f(\ell,M_ r)=2^{1/(r+1)}\ell^{(r-1)/(r+1)}(1+o(1)). \]
The author confirms this for \(r\geq 10:\)
\[ f(\ell,M_ r)=2^{1/(r+1)}\ell^{(r- 1)/(r+1)}(1+O(1/\ell^{\alpha})) \]
for any \(\alpha\), \(0<\alpha <1/(6r+6).\)
(3) \(M=\{m)\), a set of one element. Alon conjectured that if \(\ell^{1.1}\leq m\leq \ell^{1.9}\), then \(f(\ell,M)=(\ell /\bar m)(1+o(1))\), \(\ell \to \infty\), where \(\bar m\) is the smallest integer which does not divide m. The author proves that if \(c_ 1 \ell (\log \ell)^ 6<m<\ell^{3/2}/(\log \ell)^ 3\), then \(f(\ell,M)=(\ell /\bar m)+h\) where \(h=c_ 2(\ell /\bar m)\log \bar m/(\log \ell)^ 2\). Here \(c_ 1\), \(c_ 2\) are constants.
The author also proves three other auxiliary theorems using the analytical methods of Erdős-Freiman on associated problems (to appear).