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(This is not a full answer, but rather a comment that got too long)

I think https://people.kth.se/~parissis/PhD_thesis/thesis.pdf has some useful insights for your question. In particular, when $\alpha,\beta$ are integers, it works well.

I did not do the computations, but I would expect that, with some modifications, it could work for your original question about monomials. The key insight is that, if one denotes the phase $\lambda_1 x^{\alpha} + \lambda_1 x^{\beta} = \phi_{\lambda}(x),$ then we just have to find a good bound on the measure of the set $$ \{ x \in (a,b) : |\phi_{\lambda}(x)| < \alpha \}, $$ and prove that the complement has 'boundedly many' connected components. Then one uses 'regular van der Corput' on the complement, and bounds the integral by $1$ in the set above.

For the aforementioned bound on the measure of the set, maybe Lemma 2.10 in the link I sent can be useful. For the bound on the amount of components of the complement, one basically needs to count roots of a polynomial equation in the integer case, and I would assume this would not be too hard to generalize.

Notice that in the case where one has monomials - that is, in the setting of your original question - one may perform such a procedure such that the resulting constant does not depend on the interval. For generic functions $\{f_k\}$, however, these bounds would yield a $C$ which, in principle, depends on the interval under consideration. It seems that this is not relevant in the statement of your question by what I explained in the previous sentence, but in case one needs $C$ independent, then one might need to dive deeper into the link I sent...

In any case, I hope this helps :)

(This is not a full answer, but rather a comment that got too long)

I think https://people.kth.se/~parissis/PhD_thesis/thesis.pdf has some useful insights for your question. In particular, when $\alpha,\beta$ are integers, it works well.

I did not do the computations, but I would expect that, with some modifications, it could work for your original question about monomials. The key insight is that, if one denotes the phase $\lambda_1 x^{\alpha} + \lambda_1 x^{\beta} = \phi_{\lambda}(x),$ then we just have to find a good bound on the measure of the set $$ \{ x \in (a,b) : |\phi_{\lambda}(x)| < \alpha \}, $$ and prove that the complement has 'boundedly many' connected components. Then one uses 'regular van der Corput' on the complement, and bounds the integral by $1$ in the set above.

For the aforementioned bound on the measure of the set, maybe Lemma 2.10 in the link I sent can be useful. For the bound on the amount of components of the complement, one basically needs to count roots of a polynomial equation in the integer case, and I would assume this would not be too hard to generalize.

Notice that these bounds would yield a $C$ which, in principle, depends on the interval under consideration. It seems that this is not relevant in the statement of your question by what I explained in the previous sentence, but in case one needs $C$ independent - or the optimal dependency on the interval - then one might need to dive deeper into the link I sent...

In any case, I hope this helps :)

(This is not a full answer, but rather a comment that got too long)

I think https://people.kth.se/~parissis/PhD_thesis/thesis.pdf has some useful insights for your question. In particular, when $\alpha,\beta$ are integers, it works well.

I did not do the computations, but I would expect that, with some modifications, it could work for your original question about monomials. The key insight is that, if one denotes the phase $\lambda_1 x^{\alpha} + \lambda_1 x^{\beta} = \phi_{\lambda}(x),$ then we just have to find a good bound on the measure of the set $$ \{ x \in (a,b) : |\phi_{\lambda}(x)| < \alpha \}, $$ and prove that the complement has 'boundedly many' connected components. Then one uses 'regular van der Corput' on the complement, and bounds the integral by $1$ in the set above.

For the aforementioned bound on the measure of the set, maybe Lemma 2.10 in the link I sent can be useful. For the bound on the amount of components of the complement, one basically needs to count roots of a polynomial equation in the integer case, and I would assume this would not be too hard to generalize.

Notice that these bounds would yield a $C$ which, in principle, depends on the interval under consideration. It seems that this is not important in the statement of your question, but in case one needs $C$ independent, then one might need to dive deeper into the link I sent...

In any case, I hope this helps :)

(This is not a full answer, but rather a comment that got too long)

I think https://people.kth.se/~parissis/PhD_thesis/thesis.pdf has some useful insights for your question. In particular, when $\alpha,\beta$ are integers, it works well.

I did not do the computations, but I would expect that, with some modifications, it could work for your original question about monomials. The key insight is that, if one denotes the phase $\lambda_1 x^{\alpha} + \lambda_1 x^{\beta} = \phi_{\lambda}(x),$ then we just have to find a good bound on the measure of the set $$ \{ x \in (a,b) : |\phi_{\lambda}(x)| < \alpha \}, $$ and prove that the complement has 'boundedly many' connected components. Then one uses 'regular van der Corput' on the complement, and bounds the integral by $1$ in the set above.

For the aforementioned bound on the measure of the set, maybe Lemma 2.10 in the link I sent can be useful. For the bound on the amount of components of the complement, one basically needs to count roots of a polynomial equation in the integer case, and I would assume this would not be too hard to generalize.

Notice that in the case where one has monomials - that is, in the setting of your original question - one may perform such a procedure such that the resulting constant does not depend on the interval. For generic functions $\{f_k\}$, however, these bounds would yield a $C$ which, in principle, depends on the interval under consideration. It seems that this is not relevant in the statement of your question by what I explained in the previous sentence, but in case one needs $C$ independent, then one might need to dive deeper into the link I sent...

In any case, I hope this helps :)

(This is not a full answer, but rather a comment that got too long)

I think https://people.kth.se/~parissis/PhD_thesis/thesis.pdf has some useful insights for your question. In particular, when $\alpha,\beta$ are integers, it works well.

I did not do the computations, but I would expect that, with some modifications, it could work for your original question about monomials. The key insight is that, if one denotes the phase $\lambda_1 x^{\alpha} + \lambda_1 x^{\beta} = \phi_{\lambda}(x),$ then we just have to find a good bound on the measure of the set $$ \{ x \in (a,b) : |\phi_{\lambda}(x)| < \alpha \}, $$ and prove that the complement has 'boundedly many' connected components. For the aforementioned bound on the measure of the set, maybe Lemma 2.10 in the link I sent can be useful. For the bound on the amount of components of the complement, one basically needs to count roots of a polynomial equation in the integer case, and I would assume this would not be too hard to generalize.

Notice that these bounds would yield a $C$ which, in principle, depends on the interval under consideration. It seems that this is not important in the statement of your question, but in case one needs $C$ independent, then one might need to dive deeper into the link I sent...

In any case, I hope this helps :)

(This is not a full answer, but rather a comment that got too long)

I think https://people.kth.se/~parissis/PhD_thesis/thesis.pdf has some useful insights for your question. In particular, when $\alpha,\beta$ are integers, it works well.

I did not do the computations, but I would expect that, with some modifications, it could work for your original question about monomials. The key insight is that, if one denotes the phase $\lambda_1 x^{\alpha} + \lambda_1 x^{\beta} = \phi_{\lambda}(x),$ then we just have to find a good bound on the measure of the set $$ \{ x \in (a,b) : |\phi_{\lambda}(x)| < \alpha \}, $$ and prove that the complement has 'boundedly many' connected components. Then one uses 'regular van der Corput' on the complement, and bounds the integral by $1$ in the set above.

For the aforementioned bound on the measure of the set, maybe Lemma 2.10 in the link I sent can be useful. For the bound on the amount of components of the complement, one basically needs to count roots of a polynomial equation in the integer case, and I would assume this would not be too hard to generalize.

Notice that these bounds would yield a $C$ which, in principle, depends on the interval under consideration. It seems that this is not important in the statement of your question, but in case one needs $C$ independent, then one might need to dive deeper into the link I sent...

In any case, I hope this helps :)