Diophantine problems in variables restricted to the values 0 and 1
Let \(G\) be a finite additive group. Define \(s_1(G)\) as the smallest positive integer such that, for any sequence \(g_1,\ldots, g_s\) in \(G\) with \(s>s_1(G)\), there is a nontrivial solution of \[ \varepsilon_1g_1 + \cdots + \varepsilon_sg_s =0 \] with each \(\varepsilon_i =0\text{ or }1\). The value of \(s_1(G)\) is known if \(G\) is a \(p\)-group, \(G = C_{p^{e_1}} \oplus\cdots \oplus C_{p^{e_r}}\) \((C_n\) is cyclic of order \(m)\). In this case, \(\displaystyle s_1(G) = (\sum_{j=1}^r p^{e_j} -1)\) [\textit{J.~E. Olson}, J. Number Theory 1, 8--10 (1969; Zbl 0169.02003); ibid. 1, 195--199 (1969; Zbl 0167.28004)]. More generally, \(s_d(G)\) is defined as follows. For any form \(f(x)= f(x_1,\ldots,x_s)\) with coefficients in \(G\), of degree \(d\) and with \(s > s_d(G)\), the equation \[ f(\varepsilon_1, \ldots, \varepsilon_s) =0 \] has a solution of the above type. (And \(s_d(G)\) is as small as possible!) Using group rings, the authors prove that \(s_d(G)=ds_1(G)\) for a \(p\)-group. Since the paper appeared it has come to their attention that \textit{S. Schanuel} obtained this result in 1974 [J. Number Theory 6, 284--290 (1974; Zbl 0292.10042)] by a quite different method. Using character sums (a `uniform distribution' approach) the authors obtain new estimates for \(s_1(G)\) when \(G\) is not a \(p\)-group, for instance \[ s_1(C_m^k) < c_1(\varepsilon)(m^2k)^{1 +\varepsilon} \] where \(C_m^k\) denotes the direct sum of \(k\) copies of \(C_m\). They deduce estimates for \(s_d(G)\) where \(d\ge 2\) (for instance, \(s_2(C_m) < c_2(\varepsilon) m^{4+\varepsilon}\). Now suppose the form \(f\) has real coefficients. The authors consider the problem of making \(f(\varepsilon_1, \ldots, \varepsilon_s)\) close to an integer. If \(f\) has odd degree and a bound \(K\) on the coefficients is given, they also show that \(\vert f(\underline x)\vert < 1\) where \(\underline x\) has coordinates \(0,1 \text{ or }-1\), provided \(s \ge c_3(d,K)\). This last result is an application of Ramsey's theorem.
scientific article; zbMATH DE number 3693360